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Level 1 Exponents

https://www.youtube.com/watch?v=8htcZca0JIA

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WEBVTT Kind: captions Language: en 00:00:01.200 --> 00:00:04.430 Welcome to the presentation on exponents. 00:00:04.696 --> 00:00:07.012 If I were to ask you what 00:00:07.012 --> 00:00:11.570 -- let me make sure this pen is the right width-- 00:00:11.570 --> 00:00:14.860 if I were to ask you what two times three is, 00:00:14.860 --> 00:00:17.110 well I think at this point this should be pretty easy for you. 00:00:17.120 --> 00:00:25.190 That's the same thing as two plus two plus two, which equals six. 00:00:25.190 --> 00:00:26.840 You didn't have to do this. 00:00:26.850 --> 00:00:30.595 You all know that two times three is equal to six. 00:00:30.595 --> 00:00:32.465 So what we're going to do now is 00:00:32.465 --> 00:00:37.136 we're going to learn exponents which are the same thing 00:00:37.136 --> 00:00:41.730 that multiplication is to addition, exponents are to multiplication. 00:00:41.740 --> 00:00:43.163 I'll explain that in a second. 00:00:43.163 --> 00:00:44.740 I know that probably confused you. 00:00:44.740 --> 00:00:50.380 If I were to say what two to the third power is, 00:00:50.380 --> 00:00:52.709 instead of two plus two plus two, 00:00:52.709 --> 00:01:02.070 this is equal to two times two times two, which equals eight. 00:01:02.070 --> 00:01:07.640 Or if I were to say three the second power, 00:01:07.650 --> 00:01:13.160 that is equal to three times three. 00:01:13.344 --> 00:01:19.870 Remember, I said three times two would be three plus three. 00:01:19.870 --> 00:01:23.620 So the first three times three this equals nine, 00:01:23.620 --> 00:01:25.940 and three plus three equals six. 00:01:26.000 --> 00:01:28.165 The reason why I'm doing this is because there's always this temptation 00:01:28.165 --> 00:01:30.504 when you first learn exponents is to multiply. 00:01:30.504 --> 00:01:33.812 When I first learned it and I saw three to the second power or three squared, 00:01:33.812 --> 00:01:35.631 I'd always be like oh, that's six. 00:01:35.631 --> 00:01:37.898 But you also remember it's three times three, 00:01:37.898 --> 00:01:40.035 which equals nine. 00:01:40.035 --> 00:01:42.640 Let's do some more problems. 00:01:44.930 --> 00:01:52.710 If I were to tell you negative four squared. 00:01:52.730 --> 00:01:54.969 Once again, that's the same thing as 00:01:54.969 --> 00:02:01.250 negative four times negative four. 00:02:01.260 --> 00:02:04.540 Well, we all learn from multiplying negative numbers 00:02:04.540 --> 00:02:07.540 that a negative times a negative is a positive. 00:02:07.540 --> 00:02:08.861 And then four times four, 00:02:08.861 --> 00:02:10.029 well this equals positive sixteen. 00:02:10.029 --> 00:02:11.584 You don't have to write the positive, 00:02:11.584 --> 00:02:13.369 I'm just doing that for emphasis. 00:02:13.369 --> 00:02:19.584 If I would ask you what negative four to the third power is, 00:02:19.584 --> 00:02:26.440 well that equals negative four times negative four 00:02:26.440 --> 00:02:29.620 times negative four. 00:02:29.620 --> 00:02:32.340 Well we know already that negative four times negative four, 00:02:32.350 --> 00:02:35.130 that this equals sixteen, positive sixteen, 00:02:35.130 --> 00:02:38.470 and then we multiply that times negative four. 00:02:38.470 --> 00:02:42.490 And then that equals minus sixty-four. 00:02:42.500 --> 00:02:44.720 So something very interesting here to observe. 00:02:44.720 --> 00:02:46.267 When I took a negative number, 00:02:46.267 --> 00:02:47.968 and we call this the base, 00:02:47.968 --> 00:02:49.134 when the base is negative, 00:02:49.134 --> 00:02:50.454 in this case, negative four, 00:02:50.681 --> 00:02:53.248 and I raise it to an even power, 00:02:53.248 --> 00:02:55.552 I got a positive number, right? 00:02:55.552 --> 00:02:58.553 Negative four to an even power is positive sixteen, 00:02:58.553 --> 00:03:02.600 and when I took a negative number to an odd power, to three, 00:03:02.610 --> 00:03:03.984 I got a negative number. 00:03:03.984 --> 00:03:06.332 And that makes sense because every time 00:03:06.332 --> 00:03:08.298 you multiply by a negative number again, 00:03:08.298 --> 00:03:09.749 it switches signs. 00:03:09.749 --> 00:03:12.189 I'll show you the, I guess you'd call it 00:03:12.189 --> 00:03:13.939 the simplest example. 00:03:13.939 --> 00:03:18.580 Negative one to the one power is equal to negative one, right? 00:03:18.590 --> 00:03:21.930 Because that's just negative one times itself one time. 00:03:21.930 --> 00:03:26.877 And if I said negative one squared, 00:03:26.877 --> 00:03:29.435 well that's negative one times negative one, 00:03:29.435 --> 00:03:32.418 well that equals positive one. 00:03:32.418 --> 00:03:39.584 But if I said negative one to the third power, 00:03:39.584 --> 00:03:44.750 once again that's negative one times negative one times negative one. 00:03:44.750 --> 00:03:47.850 Well now this equals negative one times negative one is positive one 00:03:47.860 --> 00:03:50.340 times negative one equals negative one again. 00:03:50.340 --> 00:03:51.803 So I could tell you 00:03:51.803 --> 00:03:56.070 what negative one to the fifty-first power is. 00:03:56.080 --> 00:03:57.892 Because fifty-one is odd, 00:03:57.892 --> 00:04:01.322 we know that that is equal to negative one. 00:04:01.322 --> 00:04:03.250 If it was a fifty then it would be a positive one. 00:04:03.468 --> 00:04:05.510 Hope I didn't confuse you too much. 00:04:05.520 --> 00:04:08.932 Let's do a couple more problems. 00:04:08.932 --> 00:04:13.059 If I asked you what five to the third power is, 00:04:13.059 --> 00:04:19.207 well that will go into five times five times five, 00:04:19.207 --> 00:04:22.601 which equals one hundred and twenty-five. 00:04:22.601 --> 00:04:23.935 So really if I were ask to ask 00:04:23.935 --> 00:04:27.703 what negative five to the third power is, 00:04:27.703 --> 00:04:36.326 that would be negative five times negative five times negative five, 00:04:36.326 --> 00:04:39.910 which would be negative one hundred and twenty-five. 00:04:39.920 --> 00:04:41.686 Now one principle of exponents 00:04:41.686 --> 00:04:45.060 that might not seem completely intuitive to you at first 00:04:45.060 --> 00:04:46.869 is when I raise something to the zero power. 00:04:46.869 --> 00:04:51.111 So let's say I had two to the zero power. 00:04:51.111 --> 00:04:54.200 It turns out that anything to the zero power is equal to one. 00:04:54.200 --> 00:04:56.442 So two to the zero power is one, 00:04:56.442 --> 00:05:00.384 three to the zero power is equal to one, 00:05:00.384 --> 00:05:07.471 negative nine hundred to the zero power is equal to one. 00:05:07.471 --> 00:05:09.571 And let me see if I can give you a little bit of intuition of 00:05:09.571 --> 00:05:13.009 why that is actually the case. 00:05:15.916 --> 00:05:17.981 So if I were to ask you, 00:05:17.981 --> 00:05:23.231 let's do three to the fourth power. 00:05:23.231 --> 00:05:29.265 That equals three times three times three times three, 00:05:29.265 --> 00:05:32.249 which equals eighty-one. 00:05:32.249 --> 00:05:38.200 three to the third power is equal to twenty-seven, 00:05:38.200 --> 00:05:41.132 this three times three times three. 00:05:41.132 --> 00:05:46.930 three to second power is equal to nine. 00:05:46.930 --> 00:05:52.790 three to the first power is equal to three. 00:05:52.800 --> 00:05:54.753 Now we're going to say what's three to zero power? 00:05:54.753 --> 00:05:56.357 Well we already know, I already told you the rule, 00:05:56.357 --> 00:05:58.103 anything with the zero power is equal to one, 00:05:58.103 --> 00:06:00.040 but this will hopefully give you some intuition. 00:06:00.050 --> 00:06:03.490 When we went from the fourth power to the third power, 00:06:03.500 --> 00:06:05.640 we divided by three, right? 00:06:05.640 --> 00:06:08.240 eighty-one divided by three is twenty-seven. 00:06:08.250 --> 00:06:11.048 When you went from the third power to the second power, 00:06:11.048 --> 00:06:13.490 we divided by three. 00:06:13.490 --> 00:06:15.230 When we went from the second power from nine to three, 00:06:15.230 --> 00:06:17.350 we divided by three. 00:06:17.360 --> 00:06:19.744 So it kind of makes logical sense that 00:06:19.744 --> 00:06:22.478 when we go from the first power to the zero power, 00:06:22.478 --> 00:06:25.911 we'll just divide by three again. 00:06:25.911 --> 00:06:27.890 So three divided by three is one. 00:06:27.890 --> 00:06:30.032 Hopefully that gives you a little bit of an intuitive sense. 00:06:30.032 --> 00:06:33.480 You might want to replay that and think about why that is. 00:06:33.480 --> 00:06:36.270 And there's actually other aspects of exponents that 00:06:36.270 --> 00:06:37.719 why this also makes sense, 00:06:37.719 --> 00:06:39.683 why something to the zero power 00:06:39.683 --> 00:06:41.202 is equal to one. 00:06:41.202 --> 00:06:42.960 But let's just do some more problems in the time we have. 00:06:42.970 --> 00:06:46.750 I don't want to get you too confused. 00:06:46.750 --> 00:06:49.600 So if I were to ask you seven squared, 00:06:49.610 --> 00:06:53.720 well that's seven times seven, that's forty-nine. 00:06:53.797 --> 00:07:00.789 If I asked you negative six to the third power 00:07:00.820 --> 00:07:04.419 -- parentheses around here so you know it's a whole negative six to the third power-- 00:07:04.419 --> 00:07:09.086 that equals negative six times negative six times negative six. 00:07:09.117 --> 00:07:15.319 Negative six times negative six is positive thirty-six times negative six, 00:07:15.381 --> 00:07:16.730 and that equals what? 00:07:16.730 --> 00:07:20.127 It's one hundred and eighty and thirty-six, 00:07:20.127 --> 00:07:22.164 that's minus two hundred and sixteen, 00:07:22.164 --> 00:07:24.296 if my mental math is correct. 00:07:24.296 --> 00:07:27.332 You could have actually multiplied it out. 00:07:27.594 --> 00:07:31.200 I think you're getting the point at this point. 00:07:31.200 --> 00:07:32.344 Oh, and another thing, 00:07:32.344 --> 00:07:35.176 if I told you zero to the hundredth power, 00:07:35.176 --> 00:07:36.290 well, that's pretty easy. 00:07:36.290 --> 00:07:39.021 That's zero times itself one hundred times, 00:07:39.021 --> 00:07:41.940 which is still equal to zero. 00:07:41.940 --> 00:07:49.340 If I were to ask you one to the thousandth power, 00:07:49.350 --> 00:07:50.370 well that's just equal to one, right? 00:07:50.370 --> 00:07:52.930 You can multiply one by itself as many times as you want 00:07:52.940 --> 00:07:54.750 and you're still going to get one. 00:07:54.760 --> 00:07:59.430 And remember, if I had negative one to the one thousandth power, 00:07:59.430 --> 00:08:02.910 well, this is an even exponent so you're still going to get one. 00:08:02.920 --> 00:08:06.960 If it was negative one to the one thousand and one, 00:08:06.960 --> 00:08:07.870 then it would be negative one. 00:08:07.870 --> 00:08:09.672 I think you remember why this is, 00:08:09.672 --> 00:08:12.540 because when you multiply a negative times itself an even number of times, 00:08:12.550 --> 00:08:13.550 the negatives cancel out. 00:08:13.550 --> 00:08:16.445 And then if you multiply it by a negative one more time 00:08:16.445 --> 00:08:17.970 it becomes a negative number again. 00:08:17.980 --> 00:08:19.430 Well let's just do some normal problems. 00:08:19.430 --> 00:08:22.699 I just want to make sure you get the basics of exponents down. 00:08:23.010 --> 00:08:25.682 If I were to tell you 00:08:25.682 --> 00:08:28.132 -- let me think of a good one-- 00:08:28.132 --> 00:08:30.330 eight squared, 00:08:30.330 --> 00:08:34.947 that equals eight times eight equals sixty-four. 00:08:34.947 --> 00:08:39.442 If I were to tell you twenty-five squared, 00:08:39.442 --> 00:08:46.880 that's twenty-five times twenty-five, which equals six hundred and twenty-five. 00:08:46.880 --> 00:08:49.870 Powers of two is always very interesting. 00:08:49.880 --> 00:08:51.639 It's especially interesting 00:08:51.639 --> 00:08:53.860 if you one day go into computer science. 00:08:53.870 --> 00:08:55.496 So two to the fourth power, 00:08:55.496 --> 00:08:59.820 that's two times two times two times two. 00:08:59.820 --> 00:09:06.750 So two times two is four so this equals sixteen. 00:09:06.750 --> 00:09:08.350 And I did something very interesting here 00:09:08.360 --> 00:09:09.720 kind of on purpose. 00:09:09.720 --> 00:09:18.210 Notice that two to the fourth is equal to four times four, right? 00:09:18.210 --> 00:09:20.370 Because we did four times four here. 00:09:20.370 --> 00:09:22.678 I'm going to detail this more later on, 00:09:22.678 --> 00:09:24.020 but I want to think about what that means. 00:09:24.020 --> 00:09:27.070 Because four itself is the same thing as two squared. 00:09:27.080 --> 00:09:29.094 So we learned, just real fast, 00:09:29.094 --> 00:09:30.381 two to the fourth is the same thing 00:09:30.381 --> 00:09:34.130 as two squared times two squared. 00:09:34.130 --> 00:09:35.742 So I'll let you sit and think about that, 00:09:35.742 --> 00:09:37.727 but other than that I think you have the general idea 00:09:37.727 --> 00:09:40.430 of how basic exponents work, 00:09:40.468 --> 00:09:43.891 and I think you're ready to try the level one exponent module. 00:09:43.891 --> 99:59:59.999 have fun.

Speed translation

https://www.youtube.com/watch?v=aTjNDKlz8G4

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en

WEBVTT Kind: captions Language: en 00:00:01.280 --> 00:00:04.450 Welcome to the presentation on units. 00:00:04.450 --> 00:00:05.570 Let's get started. 00:00:05.570 --> 00:00:09.180 So if I were to tell you -- let me make sure my pen is set up 00:00:09.180 --> 00:00:13.700 right -- if I were to tell you that someone is, let's say 00:00:13.700 --> 00:00:16.893 they're driving at a speed of -- let's say it's Zack. 00:00:21.750 --> 00:00:24.170 So let's say I have Zack. 00:00:24.170 --> 00:00:29.100 And they're driving at a speed of, let me say, 00:00:29.100 --> 00:00:38.680 28 feet per minute. 00:00:38.680 --> 00:00:44.430 So what I'm going to ask you is if he's going 28 feet in every 00:00:44.430 --> 00:00:55.890 minute, how many inches will Zack travel in 1 second? 00:00:55.890 --> 00:00:59.460 So how many inches per second is he going to be going? 00:00:59.460 --> 00:01:01.570 Let's try to figure this one out. 00:01:01.570 --> 00:01:06.700 So let's say if I had 28, and I'll write ft short for feet, 00:01:06.700 --> 00:01:11.210 feet per minute, and I'll write min short for a minute. 00:01:11.210 --> 00:01:14.330 So 28 feet per minute, let's first figure out how many 00:01:14.330 --> 00:01:16.770 inches per minute that is. 00:01:16.770 --> 00:01:28.010 Well, we know that there are 12 inches per foot, right? 00:01:28.010 --> 00:01:29.790 If you didn't know that you do now. 00:01:29.790 --> 00:01:32.590 So we know that there are 12 inches per foot. 00:01:32.590 --> 00:01:35.800 So if you're going 28 feet per minute, he's going to be going 00:01:35.800 --> 00:01:40.800 12 times that many inches per minute. 00:01:40.800 --> 00:01:47.120 So, 12 times 28 -- let me do the little work down here -- 28 00:01:47.120 --> 00:01:54.920 times 12 is 16, 56 into 280. 00:01:54.920 --> 00:01:56.930 I probably shouldn't be doing it this messy. 00:01:56.930 --> 00:01:59.620 And this kind of stuff it would be OK to use a calculator, 00:01:59.620 --> 00:02:01.260 although it's always good to do the math yourself, 00:02:01.260 --> 00:02:02.510 it's good practice. 00:02:02.510 --> 00:02:07.870 So that's 6, 5 plus 8 is 13. 00:02:07.870 --> 00:02:09.350 336. 00:02:09.350 --> 00:02:21.280 So that equals 336 inches per minute. 00:02:21.280 --> 00:02:23.420 And something interesting happened here is that you 00:02:23.420 --> 00:02:26.080 noticed that I had a foot in the numerator here, and I had a 00:02:26.080 --> 00:02:27.730 foot in the denominator here. 00:02:27.730 --> 00:02:29.890 So you can actually treat units just the same way 00:02:29.890 --> 00:02:32.730 that you would treat actual numbers or variables. 00:02:32.730 --> 00:02:34.550 You have the same number in the numerator and you have the same 00:02:34.550 --> 00:02:37.080 number in the denominator, and your multiplying not adding, 00:02:37.080 --> 00:02:38.150 you can cancel them out. 00:02:38.150 --> 00:02:41.010 So the feet and the feet canceled out and that's 00:02:41.010 --> 00:02:44.020 why we were left with inches per minute. 00:02:44.020 --> 00:02:53.730 I could have also written this as 336 foot per minute 00:02:53.730 --> 00:03:00.380 times inches per foot. 00:03:00.380 --> 00:03:03.180 Because the foot per minute came from here, and the inches 00:03:03.180 --> 00:03:05.280 per foot came from here. 00:03:05.280 --> 00:03:07.580 Then I'll just cancel this out and I would have 00:03:07.580 --> 00:03:09.000 gotten inches per minute. 00:03:09.000 --> 00:03:10.620 So anyway, I don't want to confuse you too much 00:03:10.620 --> 00:03:12.710 with all of that unit cancellation stuff. 00:03:12.710 --> 00:03:15.650 The bottom line is you just remember, well if I'm going 28 00:03:15.650 --> 00:03:18.930 feet per minute, I'm going to go 12 times that many inches 00:03:18.930 --> 00:03:22.580 per minute, right, because there are 12 inches per foot. 00:03:22.580 --> 00:03:28.380 So I'm going 336 inches per minute. 00:03:28.380 --> 00:03:30.940 So now I have the question, but we're not done, because the 00:03:30.940 --> 00:03:33.230 question is how many inches am I going to be traveling 00:03:33.230 --> 00:03:35.030 in 1 second. 00:03:35.030 --> 00:03:37.320 So let me erase some of the stuff here at the bottom. 00:03:57.170 --> 00:04:09.820 So 336 inches -- let's write it like that -- inches per minute, 00:04:09.820 --> 00:04:13.390 and I want to know how many inches per second. 00:04:13.390 --> 00:04:15.010 Well what do we know? 00:04:15.010 --> 00:04:21.730 We know that 1 minute -- and notice, I write it in the 00:04:21.730 --> 00:04:23.790 numerator here because I want to cancel it out with 00:04:23.790 --> 00:04:24.970 this minute here. 00:04:24.970 --> 00:04:27.760 1 minute is equal to how many seconds? 00:04:27.760 --> 00:04:30.370 It equals 60 seconds. 00:04:32.890 --> 00:04:37.640 And this part can be confusing, but it's always good to just 00:04:37.640 --> 00:04:39.360 take a step back and think about what I'm doing. 00:04:39.360 --> 00:04:45.370 If I'm going to be going 336 inches per minute, how many 00:04:45.370 --> 00:04:47.380 inches am I going to travel in 1 second? 00:04:47.380 --> 00:04:50.235 Am I going to travel more than 336 or am I going 00:04:50.235 --> 00:04:53.640 to travel less than 336 inches per second. 00:04:53.640 --> 00:04:55.650 Well obviously less, because a second is a much 00:04:55.650 --> 00:04:57.400 shorter period of time. 00:04:57.400 --> 00:04:59.475 So if I'm in a much shorter period of time, I'm going 00:04:59.475 --> 00:05:02.140 to be traveling a much shorter distance, if I'm 00:05:02.140 --> 00:05:03.360 going the same speed. 00:05:03.360 --> 00:05:05.910 So I should be dividing by a number, which makes sense. 00:05:05.910 --> 00:05:08.090 I'm going to be dividing by 60. 00:05:08.090 --> 00:05:10.020 I know this can be very confusing at the beginning, but 00:05:10.020 --> 00:05:12.840 that's why I always want you to think about should I be getting 00:05:12.840 --> 00:05:15.370 a larger number or should I be getting a smaller number and 00:05:15.370 --> 00:05:17.800 that will always give you a good reality check. 00:05:17.800 --> 00:05:19.830 And if you just want to look at how it turns out in terms of 00:05:19.830 --> 00:05:23.390 units, we know from the problem that we want this minutes to 00:05:23.390 --> 00:05:25.630 cancel out with something and get into seconds. 00:05:25.630 --> 00:05:28.890 So if we have minutes in the denominator in the units here, 00:05:28.890 --> 00:05:31.910 we want the minutes in the numerator here, and the seconds 00:05:31.910 --> 00:05:33.250 in the denominator here. 00:05:33.250 --> 00:05:36.950 And 1 minute is equal to 60 seconds. 00:05:36.950 --> 00:05:39.540 So here, once again, the minutes and the 00:05:39.540 --> 00:05:41.090 minutes cancel out. 00:05:41.090 --> 00:05:51.275 And we get 336 over 60 inches per second. 00:05:57.020 --> 00:06:01.440 Now if I were to actually divide this out, actually we 00:06:01.440 --> 00:06:04.910 could just divide the numerator and the denominator by 6. 00:06:04.910 --> 00:06:08.850 6 goes into 336, what, 56 times? 00:06:08.850 --> 00:06:14.330 56 over 10, and then we can divide that again by 2. 00:06:14.330 --> 00:06:18.290 So then that gets us 28 over 5. 00:06:18.290 --> 00:06:30.150 And 28 over 5 -- let's see, 5 goes into 28 five times, 25. 00:06:34.170 --> 00:06:37.680 3, 5.6. 00:06:37.680 --> 00:06:38.775 So this equals 5.6. 00:06:41.890 --> 00:06:45.530 So I think we now just solved the problem. 00:06:45.530 --> 00:06:49.580 If Zack is going 28 feet in every minute, that's his 00:06:49.580 --> 00:06:56.240 speed, he's actually going 5.6 inches per second. 00:06:56.240 --> 00:06:58.850 Hopefully that kind of made sense. 00:06:58.850 --> 00:07:00.400 Let's try to see if we could do another one. 00:07:05.200 --> 00:07:16.750 If I'm going 91 feet per second, how many miles 00:07:16.750 --> 00:07:17.730 per hour is that? 00:07:22.860 --> 00:07:27.210 Well, 91 feet per second. 00:07:29.990 --> 00:07:32.080 If we want to say how many miles that is, should we be 00:07:32.080 --> 00:07:34.480 dividing or should we be multiplying? 00:07:34.480 --> 00:07:35.640 We should be dividing because it's going to be a 00:07:35.640 --> 00:07:37.560 smaller number of miles. 00:07:37.560 --> 00:07:44.820 We know that 1 mile is equal to -- and you might want to just 00:07:44.820 --> 00:07:47.950 memorize this -- 5,280 feet. 00:07:47.950 --> 00:07:50.430 It's actually a pretty useful number to know. 00:07:50.430 --> 00:07:54.520 And then that will actually cancel out the feet. 00:07:54.520 --> 00:07:59.150 Then we want to go from seconds to hours, right? 00:07:59.150 --> 00:08:02.760 So, if we go from seconds to hours, if I can travel 91 feet 00:08:02.760 --> 00:08:05.190 per second, how many will I travel in an hour, I'm going to 00:08:05.190 --> 00:08:07.950 be getting a larger number because an hour's a much larger 00:08:07.950 --> 00:08:09.530 period of time than a second. 00:08:09.530 --> 00:08:11.500 And how many seconds are there in an hour? 00:08:11.500 --> 00:08:14.080 Well, there are 3,600 seconds in an hour. 00:08:14.080 --> 00:08:18.870 60 seconds per minute and 60 minutes per hour. 00:08:18.870 --> 00:08:26.090 So 3,600 over 1 seconds per hour. 00:08:26.090 --> 00:08:28.880 And these seconds will cancel out. 00:08:28.880 --> 00:08:32.960 Then we're just left with, we just multiply everything out. 00:08:32.960 --> 00:08:42.030 We get in the numerator, 91 times 3,600, right? 00:08:42.030 --> 00:08:44.950 91 times 1 times 3,600. 00:08:44.950 --> 00:08:50.880 In the denominator we just have 5,280. 00:08:50.880 --> 00:08:53.750 This time around I'm actually going to use a calculator -- 00:08:53.750 --> 00:08:55.860 let me bring up the calculator just to show you that I'm 00:08:55.860 --> 00:08:57.960 using the calculator. 00:08:57.960 --> 00:09:13.180 Let's see, so if I say 91 times 3,600, that equals a huge 00:09:13.180 --> 00:09:31.160 number divided by 5,280. 00:09:31.160 --> 00:09:32.730 Let me see if I can type it. 00:09:32.730 --> 00:09:44.520 91 times 3,600 divided by 5,280 -- 62.05. 00:09:44.520 --> 00:09:52.380 So that equals 62.05 miles per hour.

Dividing fractions

https://www.youtube.com/watch?v=zQMU-lsMb3U

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https://www.youtube.com/api/timedtext?v=zQMU-lsMb3U&ei=g2eUZbn4LIu4vdIP9OCj8Aw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=130A91CBB93D1015B11AE8B1A8774293EEF798E8.BC2AD2E4C5AC611ACDB31CE09B9E159C15996A0A&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.810 --> 00:00:03.110 Welcome to the presentation on dividing fractions. 00:00:03.110 --> 00:00:04.490 Let's get started. 00:00:04.490 --> 00:00:06.640 So before I give you the intuition-- actually, I might 00:00:06.640 --> 00:00:09.340 do that in a different module-- I'm just going to show you the 00:00:09.340 --> 00:00:11.740 mechanics of how you divide a fraction. 00:00:11.740 --> 00:00:13.740 And it turns out that it's actually not much 00:00:13.740 --> 00:00:16.030 more difficult than multiplying fractions. 00:00:16.030 --> 00:00:21.410 If I were to ask you, 1/2 divided by 1/2, whenever you 00:00:21.410 --> 00:00:25.110 divide by a fraction, or actually, when you divide by 00:00:25.110 --> 00:00:29.960 any number, it's the same thing as multiplying by its inverse. 00:00:29.960 --> 00:00:36.670 So 1/2 divided by 1/2 is equal to 1/2 times 2/1. 00:00:36.670 --> 00:00:44.990 We just inverted-- inverse-- the second 1/2. 00:00:44.990 --> 00:00:47.630 And we know from the multiplication module, 1/2 00:00:47.630 --> 00:00:51.110 times 2/1, well, that's just equal to 2/2, 00:00:51.110 --> 00:00:53.560 or it's equal to 1. 00:00:53.560 --> 00:00:56.020 And that makes sense because, actually, any number divided 00:00:56.020 --> 00:00:58.750 by itself is equal to 1. 00:00:58.750 --> 00:01:03.220 1/2 divided by 1/2 is 1, just like 5 divided by 5 is 1, just 00:01:03.220 --> 00:01:05.240 like 100 divided by 100 is 1. 00:01:05.240 --> 00:01:06.850 And this isn't a new principal. 00:01:06.850 --> 00:01:08.970 Actually, you were always doing it. 00:01:08.970 --> 00:01:15.000 Think about it this way: What is 2 divided by 2? 00:01:15.000 --> 00:01:16.290 Well, you know that's 1. 00:01:16.290 --> 00:01:20.560 But isn't this also the same thing as 2 times the 00:01:20.560 --> 00:01:24.210 inverse of 2, which is 1? 00:01:24.210 --> 00:01:24.950 I'll show it to you. 00:01:24.950 --> 00:01:26.990 Actually, let me give you a couple more examples to show 00:01:26.990 --> 00:01:31.340 that dividing fractions really isn't a new concept, this whole 00:01:31.340 --> 00:01:34.840 notion of multiplying by the inverse. 00:01:34.840 --> 00:01:40.540 If I were to tell you what is 12 divided by 4? 00:01:40.540 --> 00:01:42.650 Well, we know the answer to this, but I'm going to show 00:01:42.650 --> 00:01:50.640 you that this is the same thing as 12 times 1/4. 00:01:50.640 --> 00:01:56.230 12/1 times 1/4 4 is 12/4, which is 3. 00:01:56.230 --> 00:01:59.480 And 12/4 is really just another way of writing 12 divided by 4, 00:01:59.480 --> 00:02:02.535 so it's kind of a long way of getting to the same point. 00:02:02.535 --> 00:02:04.990 But I just wanted to show you that what we're doing in this 00:02:04.990 --> 00:02:07.970 module is nothing new than what we've always been doing 00:02:07.970 --> 00:02:09.320 when we divide by a number. 00:02:09.320 --> 00:02:11.360 Division is the same thing. 00:02:11.360 --> 00:02:14.310 Dividing by a number is the same thing as multiplying by 00:02:14.310 --> 00:02:15.960 the inverse of that number. 00:02:15.960 --> 00:02:19.880 And just as a review, an inverse, if I have a number 00:02:19.880 --> 00:02:28.070 A, the inverse-- inv, short for inverse-- is 1 over A. 00:02:28.070 --> 00:02:36.290 So the inverse of 2/3 is 3/2, or the inverse of 5, because 5 00:02:36.290 --> 00:02:39.670 is the same thing as 5/1, so the inverse is 1/5. 00:02:39.670 --> 00:02:41.000 We're just flipping it. 00:02:41.000 --> 00:02:43.320 We're switching the numerator and denominator. 00:02:43.320 --> 00:02:46.475 So let's do some fraction division problems. 00:02:49.270 --> 00:02:56.340 What is 2/3 divided by 5/6? 00:02:56.340 --> 00:03:05.970 Well, we know that this is the same thing as 2/3 times 6/5, 00:03:05.970 --> 00:03:09.230 and that's equal to 12/15. 00:03:09.230 --> 00:03:14.570 We can divide the numerator and denominator by 3, that's 4/5. 00:03:14.570 --> 00:03:22.900 What is 7/8 divided by 1/4? 00:03:22.900 --> 00:03:30.520 Well, that's the same thing as 7/8 times 4/1. 00:03:30.520 --> 00:03:32.820 Remember, I just flipped this 1/4. 00:03:32.820 --> 00:03:36.840 Divide by 1/4 is the same thing as multiplying by 4/1. 00:03:36.840 --> 00:03:38.230 That's all you've got to do. 00:03:38.230 --> 00:03:39.990 And then we could use a little shortcut we learned in the 00:03:39.990 --> 00:03:41.480 multiplication module. 00:03:41.480 --> 00:03:42.950 8 divided by 4 is 2. 00:03:42.950 --> 00:03:44.800 4 divided by 4 is 1. 00:03:44.800 --> 00:03:47.450 So that equals 7/2. 00:03:47.450 --> 00:03:49.900 Or if you wanted to write that as a mixed number, this is, of 00:03:49.900 --> 00:03:51.200 course, an improper fraction. 00:03:51.200 --> 00:03:53.440 Improper fractions have a numerator larger 00:03:53.440 --> 00:03:54.830 than the denominator. 00:03:54.830 --> 00:03:58.670 If you wanted to write that as a mixed number, 2 goes into 7 00:03:58.670 --> 00:04:03.680 three times with a remainder of 1, so that's 3 and a half. 00:04:03.680 --> 00:04:04.440 You can write it either way. 00:04:04.440 --> 00:04:05.990 I tend to keep it this way because it's 00:04:05.990 --> 00:04:07.800 easier to deal with. 00:04:07.800 --> 00:04:10.130 Let's do a ton of more problems, or at least as many 00:04:10.130 --> 00:04:13.830 more as we can do in the next four or five minutes. 00:04:13.830 --> 00:04:23.850 What's negative 2/3 divided by 5/2? 00:04:23.850 --> 00:04:29.110 Once again, that's the same thing as minus 2/3-- whoops-- 00:04:29.110 --> 00:04:34.850 as minus 2/3 times what? 00:04:34.850 --> 00:04:40.110 It's times the inverse of 5/2, which is 2/5, and 00:04:40.110 --> 00:04:45.630 that equals minus 4/15. 00:04:45.630 --> 00:04:52.300 What is 3/2 divided by 1/6? 00:04:52.300 --> 00:04:59.850 Well, that's just the same thing as 3/2 times 6/1, 00:04:59.850 --> 00:05:03.000 which equals 3 and 1. 00:05:03.000 --> 00:05:09.610 We just divided the 6 by 2 and the 2 by 2, so that equals 9. 00:05:09.610 --> 00:05:11.280 I think you might be getting it now. 00:05:11.280 --> 00:05:12.950 Let's see, let's do a couple more. 00:05:12.950 --> 00:05:16.290 And, of course, you can always pause, and look at this whole 00:05:16.290 --> 00:05:19.420 presentation again, so you can get confused all over again. 00:05:19.420 --> 00:05:27.240 Let's see, let's do minus 5/7 divided by 10/3. 00:05:27.240 --> 00:05:33.880 Well, this is the same thing as minus 5/7 times 3/10. 00:05:33.880 --> 00:05:35.420 I just multiplied by the inverse. 00:05:35.420 --> 00:05:38.120 That's all I keep doing over and over again. 00:05:38.120 --> 00:05:40.180 Minus 5 times 3. 00:05:40.180 --> 00:05:42.610 Minus 15. 00:05:42.610 --> 00:05:47.350 7 times 10 is 70. 00:05:47.350 --> 00:05:49.900 If we divide the numerator and the denominator by 00:05:49.900 --> 00:05:56.050 5, we get minus 3/14. 00:05:56.050 --> 00:05:57.500 We could have also just done it here. 00:05:57.500 --> 00:05:59.890 We could have done 5, 2, and we would have gotten 00:05:59.890 --> 00:06:02.510 minus 3/14 as well. 00:06:02.510 --> 00:06:05.420 Let's do one or two more problems. 00:06:05.420 --> 00:06:06.630 I think you kind of get it, though. 00:06:09.600 --> 00:06:14.500 Let's say 1/2 divided by minus 3. 00:06:14.500 --> 00:06:14.965 Ah-ha! 00:06:14.965 --> 00:06:17.940 So what happens when you take a fraction and you divide it by 00:06:17.940 --> 00:06:19.730 a whole number or an integer? 00:06:19.730 --> 00:06:22.970 Well, we know any whole number can be written as a fraction. 00:06:22.970 --> 00:06:29.010 This is the same thing as 1/2 divided by minus 3/1. 00:06:29.010 --> 00:06:33.870 And dividing by a fraction is the same thing as multiplying 00:06:33.870 --> 00:06:37.430 by it's inverse. 00:06:37.430 --> 00:06:42.150 So the inverse of negative 3/1 is negative 1/3, and this 00:06:42.150 --> 00:06:45.200 equals negative 1/6. 00:06:45.200 --> 00:06:46.040 Let's do it the other way. 00:06:46.040 --> 00:06:51.880 What if I had negative 3 divided by 1/2? 00:06:51.880 --> 00:06:52.500 Same thing. 00:06:52.500 --> 00:07:00.370 Negative 3 is the same thing as minus 3/1 divided by 1/2, which 00:07:00.370 --> 00:07:07.940 is the same thing as minus 3/1 times 2/1, which is equal to 00:07:07.940 --> 00:07:12.010 minus 6/1, which equals minus 6. 00:07:15.810 --> 00:07:17.350 Now, let me give you a little bit of intuition 00:07:17.350 --> 00:07:19.730 of why this works. 00:07:19.730 --> 00:07:24.240 Let's say I said 2 divided by 1/3. 00:07:24.240 --> 00:07:27.650 Well, we know that this is equal to 2/1 times 00:07:27.650 --> 00:07:30.120 3/1, which equals 6. 00:07:30.120 --> 00:07:32.700 So how does 2, 1/3, and 6 relate? 00:07:32.700 --> 00:07:33.690 Well, let's look at it this way. 00:07:33.690 --> 00:07:36.930 If I had two pieces of pizza. 00:07:36.930 --> 00:07:38.660 I have two pieces of pizza. 00:07:38.660 --> 00:07:41.520 Here's my two pieces of pizza right. 00:07:41.520 --> 00:07:42.530 Two right here. 00:07:42.530 --> 00:07:45.050 So I have two pieces of pizza, and I'm going to divide 00:07:45.050 --> 00:07:48.080 them into thirds. 00:07:48.080 --> 00:07:50.600 So I'm going to divide each pizza into a third. 00:07:50.600 --> 00:07:52.860 I'll draw the little Mercedes sign. 00:07:52.860 --> 00:07:57.050 So I'm dividing each pizza into a third, right? 00:07:57.050 --> 00:07:58.210 How many pieces do I have? 00:07:58.210 --> 00:08:02.925 Let's see, 1, 2, 3, 4, 5, 6. 00:08:02.925 --> 00:08:04.800 I have 6 pieces. 00:08:04.800 --> 00:08:08.140 So you might want to sit and ponder that for a little bit, 00:08:08.140 --> 00:08:12.850 but I think it might make a little bit of sense to you. 00:08:12.850 --> 00:08:17.190 Let's do one more just to tire your brain. 00:08:17.190 --> 00:08:25.750 If I had negative 7/2 divided by 4/9-- let's pick a negative 00:08:25.750 --> 00:08:30.580 4/9-- well, that's the same thing as minus 7/2 times 00:08:30.580 --> 00:08:33.720 minus 9/4, right? 00:08:33.720 --> 00:08:37.950 I just multiplied by the inverse of negative 4/9. 00:08:37.950 --> 00:08:41.220 9 times 7 is equal to-- negative 7 times negative 00:08:41.220 --> 00:08:47.800 9 is positive 63, and 2 times 4 is 8. 00:08:47.800 --> 00:08:51.460 Hopefully, I think you have a good idea of how to divide by 00:08:51.460 --> 00:08:55.960 a fraction now, and you can try out the dividing 00:08:55.960 --> 00:08:57.310 fractions modules. 00:08:57.310 --> 00:08:58.890 Have fun!

Multiplying fractions (old)

https://www.youtube.com/watch?v=Mnu16kCRW4U

vtt

https://www.youtube.com/api/timedtext?v=Mnu16kCRW4U&ei=gmeUZYKzO7mBhcIPxpOy-Ag&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=2CDED5239C4C4D1066A86F003DC6822FD6953F71.EE6A6E3BF4C9EAD6D54E003BC90D4A03070A8F2C&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.180 --> 00:00:04.170 Welcome to the presentation on multiplying fractions. 00:00:04.170 --> 00:00:08.120 Well, I think today you'll be very happy because you'll find 00:00:08.120 --> 00:00:10.860 out that this is one of the few times where multiplying 00:00:10.860 --> 00:00:14.730 something is easier than adding it, I think, or subtracting 00:00:14.730 --> 00:00:15.630 it for that matter. 00:00:15.630 --> 00:00:17.455 And if you don't believe me, let's do some problems. 00:00:20.430 --> 00:00:29.020 Let's start with 1/2 times 1/2. 00:00:29.020 --> 00:00:32.200 So when you multiply fractions it's very straightforward. 00:00:32.200 --> 00:00:34.450 It's essentially just two separate multiplication 00:00:34.450 --> 00:00:35.360 problems. 00:00:35.360 --> 00:00:39.100 You multiply the numerators, so you get 1 times 1. 00:00:39.100 --> 00:00:43.080 And you multiply the denominators, 2 times 2. 00:00:43.080 --> 00:00:45.330 1 times 1 is 1. 00:00:45.330 --> 00:00:47.870 2 times 2 is 4. 00:00:47.870 --> 00:00:51.360 So 1/2 times 1/2 is equal to 1/4. 00:00:51.360 --> 00:00:52.140 That makes sense. 00:00:52.140 --> 00:00:58.810 That's like saying 1/2 of 1/2 is 1/4, which makes sense. 00:00:58.810 --> 00:01:00.340 What if we had negative numbers? 00:01:00.340 --> 00:01:06.880 Well, if I had 1/2 times negative 1/2 -- and when you 00:01:06.880 --> 00:01:09.280 have a negative fraction it's good ascribe the 00:01:09.280 --> 00:01:09.970 negative number. 00:01:09.970 --> 00:01:11.540 I tend to ascribe the negative number to the numerator 00:01:11.540 --> 00:01:15.110 -- negative 1 over 2. 00:01:15.110 --> 00:01:17.740 You realize that negative 1/2 is the same thing 00:01:17.740 --> 00:01:19.820 as negative 1 over 2. 00:01:19.820 --> 00:01:21.950 Hopefully that make sense. 00:01:21.950 --> 00:01:25.670 So 1/2 times negative 1/2, that's just the same thing as 1 00:01:25.670 --> 00:01:34.070 times negative 1 over 2 times 2, which equals negative 1 00:01:34.070 --> 00:01:40.300 over 4, which is the same thing as negative 1/4. 00:01:40.300 --> 00:01:42.710 What if I had different denominators, and when you're 00:01:42.710 --> 00:01:44.430 adding and subtracting fractions that tends to 00:01:44.430 --> 00:01:45.320 make things difficult. 00:01:45.320 --> 00:01:47.360 Well, it's not necessarily the case here. 00:01:47.360 --> 00:01:55.590 If I had 2/3 times 1/2, just multiply the numerators, 2 00:01:55.590 --> 00:01:59.800 times 1, and you multiply that denominators 3 times 2. 00:01:59.800 --> 00:02:04.690 So you get 2 times 1 is 2, 3 times 2 is 6. 00:02:04.690 --> 00:02:06.920 And 2 over 6 we know from equivalent fractions is 00:02:06.920 --> 00:02:09.420 the same thing as 1/3. 00:02:09.420 --> 00:02:10.510 That was an interesting problem. 00:02:10.510 --> 00:02:12.590 Let's do it again and I want to show you a little trick here. 00:02:15.360 --> 00:02:22.090 So, 2 over 3 times 1/2 -- as we said, any multiplication 00:02:22.090 --> 00:02:23.890 problem you just multiply the numerators, multiply the 00:02:23.890 --> 00:02:25.620 denominators and you have your answer. 00:02:25.620 --> 00:02:31.540 But sometimes there's a little trick here where you can divide 00:02:31.540 --> 00:02:33.660 the numerators and the denominators by a number, 00:02:33.660 --> 00:02:35.500 because you know that this is going to be the same thing as 2 00:02:35.500 --> 00:02:39.360 times 1 over 3 times 2. 00:02:39.360 --> 00:02:41.415 Which is the same thing -- I'm just switching the order on top 00:02:41.415 --> 00:02:45.910 -- as 1 times 2 over 3 times 2. 00:02:45.910 --> 00:02:48.250 All I did is I switched the order on top, because you can 00:02:48.250 --> 00:02:49.990 multiply in either direction. 00:02:49.990 --> 00:02:55.170 And that's the same thing as 1/3 times 2 over 2. 00:02:55.170 --> 00:03:00.430 Well that's just is 1/3 times 1, which is equal to 1/3. 00:03:00.430 --> 00:03:01.700 And why did I do that? 00:03:01.700 --> 00:03:05.110 Well I want to show you that these 2s, these 2s, all I did 00:03:05.110 --> 00:03:08.040 is switch the order, but at all times we had 1, 2 in the 00:03:08.040 --> 00:03:10.210 numerator raider and I had 1, 2 in the denominator. 00:03:10.210 --> 00:03:13.320 If I wanted to, and this is kind of a trick for doing 00:03:13.320 --> 00:03:16.700 multiplication really fast so you don't have to reduce the 00:03:16.700 --> 00:03:22.650 final fraction too much, you get 2/3 times 1/3 -- 00:03:22.650 --> 00:03:24.650 2/3 times 1/2, sorry. 00:03:24.650 --> 00:03:26.190 You say I have a 2 in the numerator, 2 in the 00:03:26.190 --> 00:03:30.740 denominator, let me divide them both by 2, that equals 1/3. 00:03:30.740 --> 00:03:31.680 Just a fast trick. 00:03:31.680 --> 00:03:33.380 I hope I didn't confuse you. 00:03:33.380 --> 00:03:35.190 Let's do a couple of more problems, and I'll do 00:03:35.190 --> 00:03:38.230 it both with the trick and without the trick. 00:03:38.230 --> 00:03:44.460 What if I had 3/7 times 2 over 5. 00:03:44.460 --> 00:03:49.640 Well, multiply the numerators, 3 times 2 is 6. 00:03:49.640 --> 00:03:51.820 7 times 5 is 35. 00:03:51.820 --> 00:03:53.610 That's it. 00:03:53.610 --> 00:03:55.440 Let's do some negative numbers. 00:03:55.440 --> 00:04:05.420 If I had negative 3 over 4 times 2 over 11, well, that's 00:04:05.420 --> 00:04:11.630 negative 6 over 44, which is the same thing as 00:04:11.630 --> 00:04:14.920 negative 3 over 22. 00:04:14.920 --> 00:04:16.980 And we could have done that cross-dividing trick here. 00:04:16.980 --> 00:04:18.380 Let's do it again with the cross--. 00:04:21.240 --> 00:04:23.080 Times 2 over 11. 00:04:23.080 --> 00:04:25.653 We say oh, well 2 and 4, they're both divisible by 2, so 00:04:25.653 --> 00:04:27.770 let's divide them both by 2. 00:04:27.770 --> 00:04:31.190 So 2 becomes 1, 4 becomes 2, and then our answer 00:04:31.190 --> 00:04:34.280 becomes minus 3 over 22. 00:04:34.280 --> 00:04:37.170 Negative 3 times 1 is minus is 3. 00:04:37.170 --> 00:04:40.560 2 times 11 is 22. 00:04:40.560 --> 00:04:41.470 Do another one right here. 00:04:41.470 --> 00:04:49.760 If I had negative 2/5 times minus 2/5, well, that just is 00:04:49.760 --> 00:04:54.860 equal to negative 2 times negative 2 is positive 4. 00:04:54.860 --> 00:04:56.690 It's 5 times 5 is 25. 00:04:56.690 --> 00:04:57.550 4 over 25. 00:04:57.550 --> 00:04:59.450 And that's, just remember, a negative times a negative is a 00:04:59.450 --> 00:05:02.050 positive, which makes sense. 00:05:02.050 --> 00:05:03.630 Let's just do a couple more problems since we 00:05:03.630 --> 00:05:04.330 have a lot of time. 00:05:04.330 --> 00:05:06.070 But I think you probably got this by now. 00:05:06.070 --> 00:05:08.410 You're probably realizing that multiplying fractions is a 00:05:08.410 --> 00:05:11.890 lot easier than adding or subtracting them, hopefully. 00:05:11.890 --> 00:05:13.910 I guess it's not a bad thing if you find adding or subtracting 00:05:13.910 --> 00:05:16.320 fractions easy as well. 00:05:16.320 --> 00:05:18.453 Let's do -- I'm just making up numbers now 00:05:18.453 --> 00:05:25.910 -- 2/9 times 18 over 2. 00:05:25.910 --> 00:05:28.710 Well here we could, well, we have a 2 in the numerator 00:05:28.710 --> 00:05:30.310 and a 2 in the denominator. 00:05:30.310 --> 00:05:34.530 Let's divide them both by 2, so they both become 1. 00:05:34.530 --> 00:05:37.460 And we have an 18 in the numerator and a 9 00:05:37.460 --> 00:05:38.290 in the denominator. 00:05:38.290 --> 00:05:41.090 Well they both are divisible by 9, so let's divide 00:05:41.090 --> 00:05:41.840 them both by 9. 00:05:41.840 --> 00:05:47.890 So 9 becomes a 1, and the 18 becomes a 2. 00:05:47.890 --> 00:05:52.600 So you have 1 times 2 over 1 times 1, well, that just equals 00:05:52.600 --> 00:05:54.840 2 over 1 which equals 2. 00:05:54.840 --> 00:05:56.100 That was pretty straightforward. 00:05:56.100 --> 00:05:58.650 We could have done it, I guess you could call it the hard 00:05:58.650 --> 00:06:05.940 way, if we said 2 times 2 over 9 times 18 over 2. 00:06:05.940 --> 00:06:08.680 2 times 18 is 36. 00:06:08.680 --> 00:06:11.360 9 times 2 is 18. 00:06:11.360 --> 00:06:15.090 And 36 divided by 18, and we can see 18 goes into 36 two 00:06:15.090 --> 00:06:17.190 times, that also equals 2. 00:06:17.190 --> 00:06:18.280 Either way is fine. 00:06:18.280 --> 00:06:20.650 If you don't feel comfortable doing this trick right 00:06:20.650 --> 00:06:21.780 now, you don't have to. 00:06:21.780 --> 00:06:27.710 All that does is you won't end up with huge numbers in your 00:06:27.710 --> 00:06:29.880 product that you'll have to figure out if they can 00:06:29.880 --> 00:06:32.700 be reduced further. 00:06:32.700 --> 00:06:34.585 Let's do two more problems. 00:06:38.320 --> 00:06:46.120 Minus 5 over 7 times 1 over 3. 00:06:46.120 --> 00:06:49.950 Minus 5 times 1 is minus 5. 00:06:49.950 --> 00:06:52.750 7 over 3 is 21. 00:06:52.750 --> 00:06:54.270 That's it. 00:06:54.270 --> 00:06:59.650 Let me do one with the little trick I showed you. 00:06:59.650 --> 00:07:04.660 Say I had 15, and here I think you'll see why that trick 00:07:04.660 --> 00:07:11.580 is useful, over 21 times 14 over 5. 00:07:11.580 --> 00:07:13.880 Well clearly, if we multiply this out we end up with 00:07:13.880 --> 00:07:15.090 pretty big numbers. 00:07:15.090 --> 00:07:18.840 I think 220 one 105 and you have to reduce those. 00:07:18.840 --> 00:07:19.960 It becomes a big mess. 00:07:19.960 --> 00:07:22.770 But we can see that 15 and 5 are both divisible by 5. 00:07:22.770 --> 00:07:24.560 So let's divide them both by 5. 00:07:24.560 --> 00:07:27.260 So 15 divided by 5 is 3. 00:07:27.260 --> 00:07:30.180 5 divided by 5 is 1. 00:07:30.180 --> 00:07:33.240 14 and 21, they're both divisible by 7. 00:07:33.240 --> 00:07:36.960 14 divided by 7 is 2. 00:07:36.960 --> 00:07:40.770 21 divided by 7 is 3. 00:07:40.770 --> 00:07:48.640 So we got 3 times 2 is 6 over 3 times 1 is 3, which equals 2. 00:07:48.640 --> 00:07:50.090 That's the same thing as what I said before. 00:07:50.090 --> 00:07:52.510 If we had multiplied 15 times 14 that would 00:07:52.510 --> 00:07:56.200 have been 210 I think. 00:07:56.200 --> 00:07:58.430 Yeah, 15 times 14 is 210. 00:07:58.430 --> 00:08:02.100 And 21 times 5 would have been 105, and you would have to say, 00:08:02.100 --> 00:08:05.320 I guess in this case it's kind of obvious, that 210 is 2 times 00:08:05.320 --> 00:08:07.840 105 and you would have gotten 2 as well. 00:08:07.840 --> 00:08:09.530 So hopefully I didn't confuse you too much 00:08:09.530 --> 00:08:10.670 with that last problem. 00:08:10.670 --> 00:08:12.900 But I hope you realize multiplication's pretty 00:08:12.900 --> 00:08:13.110 straightforward. 00:08:13.110 --> 00:08:15.340 You just multiply the numerators, you multiply the 00:08:15.340 --> 00:08:18.400 denominators, and then if you have to reduce you reduce, but 00:08:18.400 --> 00:08:19.500 you're pretty much done. 00:08:19.500 --> 00:08:22.050 I think you're ready now to try the multiplication module, 00:08:22.050 --> 00:08:25.120 and I hope you have fun.

Adding and subtracting fractions

https://www.youtube.com/watch?v=52ZlXsFJULI

vtt

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en

WEBVTT Kind: captions Language: en 00:00:01.610 --> 00:00:03.710 Welcome to the presentation on adding and 00:00:03.710 --> 00:00:05.300 subtracting fractions. 00:00:05.300 --> 00:00:08.450 Let's get started. 00:00:08.450 --> 00:00:12.120 Let's start with what I hope shouldn't confuse you too much. 00:00:12.120 --> 00:00:15.130 This should hopefully be a relatively easy question. 00:00:15.130 --> 00:00:23.940 If I were to ask you what 1/4 plus 1/4 is. 00:00:23.940 --> 00:00:25.270 Let's think about what that means. 00:00:25.270 --> 00:00:32.420 Let's say we had a pie and it was divided into four pieces. 00:00:32.420 --> 00:00:35.180 So this is like saying this first 1/4 right here -- let me 00:00:35.180 --> 00:00:37.730 do it in a different color. 00:00:37.730 --> 00:00:39.530 This 1/4 right here, let's say it's this 00:00:39.530 --> 00:00:42.400 1/4 of the pie, right? 00:00:42.400 --> 00:00:45.570 And we're going to add it to another 1/4 of the pie. 00:00:45.570 --> 00:00:51.610 Let's make it this one -- let me change the color -- pink. 00:00:51.610 --> 00:00:57.070 This 1/4, this pink 1/4 is this 1/4 of the pie. 00:00:57.070 --> 00:01:02.270 So if I were to eat both 1/4s or 1/4 and then I eat another 00:01:02.270 --> 00:01:04.570 1/4, how much have I eaten? 00:01:04.570 --> 00:01:08.260 Well, you could look from just the picture, I have now eaten 2 00:01:08.260 --> 00:01:10.280 out of the 4 pieces of the pie. 00:01:10.280 --> 00:01:15.480 So if I eat 1/4 of a piece of pie or 1/4 of a pie, and then I 00:01:15.480 --> 00:01:21.550 eat another 1/4 of a pie, I will have eaten 2/4 of the pie. 00:01:21.550 --> 00:01:23.940 And we know from the equivalent fractions module that this is 00:01:23.940 --> 00:01:27.490 the same thing as I've eaten 1/2 of the pie, 00:01:27.490 --> 00:01:28.330 which makes sense. 00:01:28.330 --> 00:01:32.150 If I eat 2 out of 4 pieces of a pie, then I've eaten 1/2 of it. 00:01:32.150 --> 00:01:33.700 And if we look at it mathematically, what 00:01:33.700 --> 00:01:34.960 happened here? 00:01:34.960 --> 00:01:39.230 Well the denominators or the bottom numbers, the bottom 00:01:39.230 --> 00:01:41.270 numbers in the fraction stayed the same. 00:01:41.270 --> 00:01:42.730 Because that's just the total number of pieces 00:01:42.730 --> 00:01:44.360 I have in this example. 00:01:44.360 --> 00:01:47.930 Well, I added the numerators, which makes sense. 00:01:47.930 --> 00:01:51.490 I had 1 out of the 4 pieces of pie, then I ate another 1 out 00:01:51.490 --> 00:01:54.600 of the 4 pieces of pie, so I ate 2 out of the 4 pieces 00:01:54.600 --> 00:01:56.510 of pie, which is 1/2. 00:01:56.510 --> 00:01:57.840 Let me do a couple more examples. 00:02:01.820 --> 00:02:09.240 What is 2/5 plus 1/5? 00:02:09.240 --> 00:02:11.760 Well we do the same thing here. 00:02:11.760 --> 00:02:14.230 We first check to make sure the denominators are the same -- 00:02:14.230 --> 00:02:16.130 we'll learn in a second what we do when the denominators 00:02:16.130 --> 00:02:16.900 are different. 00:02:16.900 --> 00:02:19.340 If the denominators are the same, the denominator of the 00:02:19.340 --> 00:02:21.030 answer will be the same. 00:02:21.030 --> 00:02:22.480 And we just add the numerators. 00:02:22.480 --> 00:02:27.580 2/5 plus 1/5 is just 2 plus 1 over 5, which 00:02:27.580 --> 00:02:31.100 is equal to 3 over 5. 00:02:31.100 --> 00:02:33.390 And it works the same way with subtraction. 00:02:33.390 --> 00:02:42.430 If I had 3 over 7 minus 2 over 7, that just equals 1 over 7. 00:02:42.430 --> 00:02:45.480 I just subtracted the 3, I subtracted the 2 from the 3 00:02:45.480 --> 00:02:48.150 to get 1 and I kept the denominator the same. 00:02:48.150 --> 00:02:48.860 Which makes sense. 00:02:48.860 --> 00:02:53.040 If I have 3 out of the 7 pieces of a pie and I were to give 00:02:53.040 --> 00:02:57.800 away 2 out of the 7 pieces of a pie, I'd be left with 1 of 00:02:57.800 --> 00:03:00.180 the 7 pieces of a pie. 00:03:00.180 --> 00:03:02.370 So now let's tackle -- I think it should be pretty 00:03:02.370 --> 00:03:04.330 straightforward when we have the same denominator. 00:03:04.330 --> 00:03:05.870 Remember, the denominator is just the bottom 00:03:05.870 --> 00:03:06.880 number in a fraction. 00:03:06.880 --> 00:03:08.410 Numerator is the top number. 00:03:08.410 --> 00:03:11.430 What happens when we have different denominators? 00:03:11.430 --> 00:03:15.100 Well, hopefully it won't be too difficult. 00:03:15.100 --> 00:03:24.330 Let's say I have 1/4 plus 1/2. 00:03:24.330 --> 00:03:27.190 Let's go back to that original pie example. 00:03:27.190 --> 00:03:28.260 Let me draw that pie. 00:03:33.900 --> 00:03:37.250 So this first 1/4 right here, let's just color it in, 00:03:37.250 --> 00:03:40.470 that's this 1/4 of the pie. 00:03:40.470 --> 00:03:44.550 And now I'm going to eat another 1/2 of the pie. 00:03:44.550 --> 00:03:46.460 So I'm going to eat 1/2 of the pie. 00:03:46.460 --> 00:03:49.100 So this 1/2. 00:03:49.100 --> 00:03:51.400 I'll eat this whole 1/2 of the pie. 00:03:54.290 --> 00:03:55.230 So what does that equal? 00:03:55.230 --> 00:03:57.190 Well, there's a couple of ways we could think about it. 00:03:57.190 --> 00:03:59.210 First we could just re-write 1/2. 00:03:59.210 --> 00:04:04.135 1/2 of the pie, that's actually the same thing as 2/4, right? 00:04:06.950 --> 00:04:12.290 There's 1/4 here and then another 1/4 here. 00:04:12.290 --> 00:04:15.470 So 1/2 is the same thing as 2/4, and we know that from the 00:04:15.470 --> 00:04:17.520 equivalent fractions module. 00:04:17.520 --> 00:04:21.460 So we know that 1/4 plus 1/2, this is the same thing as 00:04:21.460 --> 00:04:27.110 saying 1/4 plus 2/4, right? 00:04:27.110 --> 00:04:36.160 And all I did here is I changed the 1/2 to a 2/4 by essentially 00:04:36.160 --> 00:04:38.820 multiplying the numerator and the denominator of 00:04:38.820 --> 00:04:40.410 this fraction by 2. 00:04:40.410 --> 00:04:41.740 And you can do that to any fraction. 00:04:41.740 --> 00:04:43.890 As long as you multiply the numerator and the denominator 00:04:43.890 --> 00:04:47.620 by the same number, you can multiply by anything. 00:04:47.620 --> 00:04:52.880 That makes sense because 1/2 times 1 is equal 00:04:52.880 --> 00:04:54.440 to 1/2, you know that. 00:04:54.440 --> 00:05:00.080 Well another way of writing 1 is 1/2 times 2/2. 00:05:00.080 --> 00:05:04.490 2 over 2 is the same thing as 1, and that equals 2 over 4. 00:05:04.490 --> 00:05:07.380 The reason why I picked 2 is because I wanted to get 00:05:07.380 --> 00:05:08.580 the same denominator here. 00:05:11.230 --> 00:05:13.530 I hope I'm not completely confusing you. 00:05:13.530 --> 00:05:15.220 Well, let's just finish up this problem. 00:05:15.220 --> 00:05:19.220 So we have 1/4 plus 2/4, so we know that we just add the 00:05:19.220 --> 00:05:22.570 numerators, 3, and the denominators are the same, 3/4. 00:05:22.570 --> 00:05:25.190 And if we look at the picture, true enough, we have 00:05:25.190 --> 00:05:29.380 eaten 3/4 of this pie. 00:05:29.380 --> 00:05:30.320 Let's do another one. 00:05:34.030 --> 00:05:44.730 Let's do 1/2 plus 1/3. 00:05:44.730 --> 00:05:47.570 Well once again, we want to get both denominators to be the 00:05:47.570 --> 00:05:51.370 same, but you can't just multiply one of them to get -- 00:05:51.370 --> 00:05:53.850 there's nothing I can multiply 3 by to get 2, or there's no, 00:05:53.850 --> 00:05:56.510 at least, integer I can multiply 3 by to get 2. 00:05:56.510 --> 00:05:58.900 And there's nothing I can multiply 2 by to get 3. 00:05:58.900 --> 00:06:01.870 So I have to multiply both of them so they equal each other. 00:06:01.870 --> 00:06:06.230 It turns out that what we want for, what we'll call the common 00:06:06.230 --> 00:06:08.850 denominator, it turns out to be the least common 00:06:08.850 --> 00:06:11.130 multiple of 2 and 3. 00:06:11.130 --> 00:06:13.400 Well what's the least common multiple of 2 and 3? 00:06:13.400 --> 00:06:16.750 Well that's the smallest number that's a multiple 00:06:16.750 --> 00:06:17.860 of both 2 and 3. 00:06:17.860 --> 00:06:19.390 Well the smallest number that's a multiple of 00:06:19.390 --> 00:06:23.480 both 2 and 3 is 6. 00:06:23.480 --> 00:06:27.890 So let's convert both of these fractions to something over 6. 00:06:27.890 --> 00:06:30.340 So 1/2 is equal to what over 6. 00:06:30.340 --> 00:06:33.320 You should know this from the equivalent fractions module. 00:06:33.320 --> 00:06:37.500 Well if I eat 1/2 of a pizza with 6 pieces, I would have 00:06:37.500 --> 00:06:40.270 eaten 3 pieces, right? 00:06:40.270 --> 00:06:40.820 That make sense. 00:06:40.820 --> 00:06:43.950 1 is 1/2 of 2, 3 is 1/2 of 6. 00:06:43.950 --> 00:06:47.650 Similarly, if I eat 1/3 of a pizza with 6 pieces, it's 00:06:47.650 --> 00:06:50.730 the same thing as 2 over 6. 00:06:50.730 --> 00:06:57.700 So 1/2 plus 1/3 is the same thing as 3/6 plus 2/6. 00:06:57.700 --> 00:06:58.980 Notice I didn't do anything crazy. 00:06:58.980 --> 00:07:02.030 All I did is I re-wrote both of these fractions with 00:07:02.030 --> 00:07:03.220 different denominators. 00:07:03.220 --> 00:07:06.050 I essentially changed the number of pieces in the 00:07:06.050 --> 00:07:08.830 pie, if that helps at all. 00:07:08.830 --> 00:07:10.410 Now that we're at this point then the problem 00:07:10.410 --> 00:07:11.190 becomes very easy. 00:07:11.190 --> 00:07:14.830 We just add the numerators, 3 plus 2 is 5, and we keep 00:07:14.830 --> 00:07:16.430 the denominators the same. 00:07:16.430 --> 00:07:18.970 3 over 6 plus 2 over 6 equals 5/6. 00:07:22.240 --> 00:07:24.740 And subtraction is the same thing. 00:07:24.740 --> 00:07:31.460 1/2 minus 1/3, well that's the same thing as 3 00:07:31.460 --> 00:07:35.110 over 6 minus 2 over 6. 00:07:35.110 --> 00:07:39.530 Well that equals 1 over 6. 00:07:39.530 --> 00:07:43.060 Let's do a bunch more problems and hopefully you'll 00:07:43.060 --> 00:07:43.990 start to get it. 00:07:43.990 --> 00:07:47.120 And always remember you can re-watch the presentation, or 00:07:47.120 --> 00:07:49.630 you can pause it and try to do the problems yourself, because 00:07:49.630 --> 00:07:53.250 I think sometimes I talk fast. 00:07:53.250 --> 00:07:55.100 Let me throw you a curve ball. 00:07:55.100 --> 00:07:59.330 What's 1/10 minus 1? 00:07:59.330 --> 00:08:01.630 Well, one doesn't even look like a fraction. 00:08:01.630 --> 00:08:04.150 But you can write it as a fraction. 00:08:04.150 --> 00:08:07.970 Well that's the same thing as 1/10 minus -- how could we 00:08:07.970 --> 00:08:11.020 write 1 so it has the denominator of 10? 00:08:11.020 --> 00:08:11.590 Right. 00:08:11.590 --> 00:08:14.830 It's the same thing as 10 over 10, right? 00:08:14.830 --> 00:08:16.330 10 over 10 is 1. 00:08:16.330 --> 00:08:20.900 So 1/10 minus 10 over 10 is the same thing as 1 minus 10 -- 00:08:20.900 --> 00:08:24.950 remember, we only subtract the numerators and we keep the 00:08:24.950 --> 00:08:31.170 denominator 10, and that equals negative 9 over 10. 00:08:31.170 --> 00:08:34.380 1/10 minus 1 is equal to negative 9 over 10. 00:08:34.380 --> 00:08:35.920 Let's do another one. 00:08:35.920 --> 00:08:36.470 Let's do one more. 00:08:36.470 --> 00:08:38.680 I think that's all I have time for. 00:08:38.680 --> 00:08:47.320 Let's do minus 1/9 minus 1 over 4. 00:08:47.320 --> 00:08:53.770 Well the least common multiple of 0 and 4 is 36. 00:08:53.770 --> 00:08:55.590 So that's equal to 36. 00:08:55.590 --> 00:09:00.070 So what's negative 1/9 where we change the denominator 00:09:00.070 --> 00:09:02.010 from 9 to 36? 00:09:02.010 --> 00:09:05.030 Well, we multiply 9 times 4 to get 36. 00:09:05.030 --> 00:09:07.230 We have to multiply the numerator times 4 as well. 00:09:07.230 --> 00:09:11.860 So we have negative 1, so it becomes negative 4. 00:09:11.860 --> 00:09:16.860 Then minus over 36. 00:09:16.860 --> 00:09:20.110 Well to go from 4 to 36, we have to multiply this fraction 00:09:20.110 --> 00:09:23.070 by 9, or we have to multiply the denominator by 9, so you 00:09:23.070 --> 00:09:25.190 also have to multiply the numerator by 9. 00:09:25.190 --> 00:09:28.370 1 times 9 is 9. 00:09:28.370 --> 00:09:35.770 So this equals minus 4 minus 9 over 36, which equals 00:09:35.770 --> 00:09:39.580 minus 13 over 36. 00:09:39.580 --> 00:09:41.850 I think that's all I have time for right now, and I'll 00:09:41.850 --> 00:09:44.000 probably add a couple more modules, but I think you might 00:09:44.000 --> 00:09:47.520 be ready now to do the adding and subtracting module. 00:09:47.520 --> 00:09:49.020 Have fun.

Multiplying and dividing negative numbers

https://www.youtube.com/watch?v=d8lP5tR2R3Q

vtt

https://www.youtube.com/api/timedtext?v=d8lP5tR2R3Q&ei=g2eUZZ7_AazLp-oPs6i-oAE&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A3B3CB9CCE1D919BCD75246E3FED52AB2BB596A6.9439BECFAD68C59E1B9B6A358560A43D22C7F4D9&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.940 --> 00:00:03.530 Welcome to the presentation on multiplying and 00:00:03.530 --> 00:00:05.270 dividing negative numbers. 00:00:05.270 --> 00:00:06.750 Let's get started. 00:00:06.750 --> 00:00:08.920 I think you're going to find that multiplying and dividing 00:00:08.920 --> 00:00:10.730 negative numbers are a lot easier than it might 00:00:10.730 --> 00:00:12.000 look initially. 00:00:12.000 --> 00:00:13.760 You just have to remember a couple rules, and I'm going to 00:00:13.760 --> 00:00:15.790 teach probably in the future like I'm actually going to give 00:00:15.790 --> 00:00:18.000 you more intuition on why these rules work. 00:00:18.000 --> 00:00:20.860 But first let me just teach you the basic rules. 00:00:20.860 --> 00:00:25.220 So the basic rules are when you multiply two negative numbers, 00:00:25.220 --> 00:00:31.790 so let's say I had negative 2 times negative 2. 00:00:31.790 --> 00:00:33.970 First you just look at each of the numbers as if there 00:00:33.970 --> 00:00:35.240 was no negative sign. 00:00:35.240 --> 00:00:40.150 Well you say well, 2 times 2 that equals 4. 00:00:40.150 --> 00:00:42.670 And it turns out that if you have a negative times a 00:00:42.670 --> 00:00:45.175 negative, that that equals a positive. 00:00:45.175 --> 00:00:47.710 So let's write that first rule down. 00:00:47.710 --> 00:00:54.285 A negative times a negative equals a positive. 00:00:56.860 --> 00:01:02.400 What if it was negative 2 times positive 2? 00:01:02.400 --> 00:01:04.770 Well in this case, let's first of all look at the 00:01:04.770 --> 00:01:06.060 two numbers without signs. 00:01:06.060 --> 00:01:10.170 We know that 2 times 2 is 4. 00:01:10.170 --> 00:01:13.740 But here we have a negative times a positive 2, and it 00:01:13.740 --> 00:01:15.910 turns out that when you multiply a negative times a 00:01:15.910 --> 00:01:18.920 positive you get a negative. 00:01:18.920 --> 00:01:20.260 So that's another rule. 00:01:20.260 --> 00:01:26.255 Negative times positive is equal to negative. 00:01:28.900 --> 00:01:34.760 What happens if you have a positive 2 times a negative 2? 00:01:34.760 --> 00:01:37.140 I think you'll probably guess this one right, as you can tell 00:01:37.140 --> 00:01:41.170 that these two are pretty much the same thing by, I believe 00:01:41.170 --> 00:01:44.800 it's the transitive property -- no, no I think it's the 00:01:44.800 --> 00:01:45.860 communicative property. 00:01:45.860 --> 00:01:47.700 I have to remember that. 00:01:47.700 --> 00:01:51.780 But 2 times negative 2, this also equals negative 4. 00:01:51.780 --> 00:01:57.640 So we have the final rule that a positive times a negative 00:01:57.640 --> 00:01:58.980 also equals the negative. 00:02:01.490 --> 00:02:03.985 And actually these second two rules, they're kind 00:02:03.985 --> 00:02:04.990 of the same thing. 00:02:04.990 --> 00:02:07.570 A negative times a positive is a negative, or a positive 00:02:07.570 --> 00:02:09.140 times a negative is negative. 00:02:09.140 --> 00:02:13.730 You could also say that as when the signs are different and 00:02:13.730 --> 00:02:16.400 you multiply the two numbers, you get a negative number. 00:02:16.400 --> 00:02:18.890 And of course, you already know what happens when you have a 00:02:18.890 --> 00:02:21.550 positive times a positive. 00:02:21.550 --> 00:02:22.840 Well that's just a positive. 00:02:22.840 --> 00:02:23.900 So let's review. 00:02:23.900 --> 00:02:27.970 Negative times a negative is a positive. 00:02:27.970 --> 00:02:30.390 A negative times a positive is a negative. 00:02:30.390 --> 00:02:32.730 A positive times a negative is a negative. 00:02:32.730 --> 00:02:36.290 And positive times each other equals positive. 00:02:36.290 --> 00:02:39.980 I think that last little bit completely confused you. 00:02:39.980 --> 00:02:42.270 Maybe I can simplify it for you. 00:02:42.270 --> 00:02:46.350 What if I just told you if when you're multiplying and they're 00:02:46.350 --> 00:02:55.175 the same signs that gets you a positive result. 00:02:57.840 --> 00:03:11.010 And different signs gets you a negative result. 00:03:11.010 --> 00:03:17.780 So that would be either, let's say a 1 times 1 is equal to 1, 00:03:17.780 --> 00:03:22.120 or if I said negative 1 times negative 1 is equal to 00:03:22.120 --> 00:03:23.510 positive 1 as well. 00:03:23.510 --> 00:03:29.150 Or if I said 1 times negative 1 is equal to negative 1, or 00:03:29.150 --> 00:03:32.600 negative 1 times 1 is equal to negative 1. 00:03:32.600 --> 00:03:36.130 You see how on the bottom two problems I had two different 00:03:36.130 --> 00:03:38.590 signs, positive 1 and negative 1? 00:03:38.590 --> 00:03:41.120 And the top two problems, this one right here 00:03:41.120 --> 00:03:42.680 both 1s are positive. 00:03:42.680 --> 00:03:45.970 And this one right here both 1s are negative. 00:03:45.970 --> 00:03:49.110 So let's do a bunch of problems now, and hopefully it'll hit 00:03:49.110 --> 00:03:51.510 the point home, and you also could try to do along the 00:03:51.510 --> 00:03:54.010 practice problems and also give the hints and give you what 00:03:54.010 --> 00:03:56.000 rules to use, so that should help you as well. 00:03:58.960 --> 00:04:06.750 So if I said negative 4 times positive 3, well 4 times 00:04:06.750 --> 00:04:11.820 3 is 12, and we have a negative and a positive. 00:04:11.820 --> 00:04:15.670 So different signs mean negative. 00:04:15.670 --> 00:04:19.060 So negative 4 times 3 is a negative 12. 00:04:19.060 --> 00:04:21.310 That makes sense because we're essentially saying what's 00:04:21.310 --> 00:04:25.070 negative 4 times itself three times, so it's like negative 4 00:04:25.070 --> 00:04:27.800 plus negative 4 plus negative 4, which is negative 12. 00:04:27.800 --> 00:04:31.120 If you've seen the video on adding and subtracting negative 00:04:31.120 --> 00:04:34.200 numbers, you probably should watch first. 00:04:34.200 --> 00:04:35.210 Let's do another one. 00:04:35.210 --> 00:04:40.430 What if I said minus 2 times minus 7. 00:04:40.430 --> 00:04:42.470 And you might want to pause the video at any time to see if you 00:04:42.470 --> 00:04:44.030 know how to do it and then restart it to see 00:04:44.030 --> 00:04:45.420 what the answer is. 00:04:45.420 --> 00:04:51.190 Well, 2 times 7 is 14, and we have the same sign here, so 00:04:51.190 --> 00:04:53.530 it's a positive 14 -- normally you wouldn't have to write the 00:04:53.530 --> 00:04:56.930 positive but that makes it a little bit more explicit. 00:04:56.930 --> 00:05:05.880 And what if I had -- let me think -- 9 times negative 5. 00:05:05.880 --> 00:05:08.800 Well, 9 times 5 is 45. 00:05:08.800 --> 00:05:13.660 And once again, the signs are different so it's a negative. 00:05:13.660 --> 00:05:18.010 And then finally what if it I had -- let me think of some 00:05:18.010 --> 00:05:24.540 good numbers -- minus 6 times minus 11. 00:05:24.540 --> 00:05:29.730 Well, 6 times 11 is 66 and then it's a negative and 00:05:29.730 --> 00:05:31.720 negative, it's a positive. 00:05:31.720 --> 00:05:32.910 Let me give you a trick problem. 00:05:32.910 --> 00:05:39.100 What is 0 times negative 12? 00:05:39.100 --> 00:05:42.740 Well, you might say that the signs are different, but 00:05:42.740 --> 00:05:46.460 0 is actually neither positive nor negative. 00:05:46.460 --> 00:05:48.315 And 0 times anything is still 0. 00:05:48.315 --> 00:05:52.080 It doesn't matter if the thing you multiply it by is a 00:05:52.080 --> 00:05:53.650 negative number or a positive number. 00:05:53.650 --> 00:05:57.630 0 times anything is still 0. 00:05:57.630 --> 00:06:00.020 So let's see if we can apply these same rules to division. 00:06:00.020 --> 00:06:03.080 It actually turns out that the same rules apply. 00:06:03.080 --> 00:06:09.030 If I have 9 divided by negative 3. 00:06:09.030 --> 00:06:11.820 Well, first we say what's 9 divided by 3? 00:06:11.820 --> 00:06:13.640 Well that's 3. 00:06:13.640 --> 00:06:17.920 And they have different signs, positive 9, negative 3. 00:06:17.920 --> 00:06:22.190 So different signs means a negative. 00:06:22.190 --> 00:06:27.520 9 divided by negative 3 is equal to negative 3. 00:06:27.520 --> 00:06:33.830 What is minus 16 divided by 8? 00:06:33.830 --> 00:06:37.790 Well, once again, 16 divided by 8 is 2, but 00:06:37.790 --> 00:06:39.370 the signs are different. 00:06:39.370 --> 00:06:44.830 Negative 16 divided by positive 8, that equals negative 2. 00:06:44.830 --> 00:06:49.140 Remember, different signs will get you a negative result. 00:06:49.140 --> 00:07:00.500 What is minus 54 divided by minus 6? 00:07:00.500 --> 00:07:04.320 Well, 54 divided by 6 is 9. 00:07:04.320 --> 00:07:09.050 And since both terms, the divisor and the dividend, are 00:07:09.050 --> 00:07:13.890 both negative -- negative 54 and negative 6 -- it turns out 00:07:13.890 --> 00:07:15.000 that the answer is positive. 00:07:15.000 --> 00:07:19.310 Remember, same signs result in a positive quotient in this 00:07:19.310 --> 00:07:22.350 example we did before, it was product. 00:07:22.350 --> 00:07:24.730 Let's do one more. 00:07:24.730 --> 00:07:30.500 Obviously, 0 divided by anything is still 0. 00:07:30.500 --> 00:07:31.510 That's pretty straightforward. 00:07:31.510 --> 00:07:33.200 And of course, you can't divide anything by 0 00:07:33.200 --> 00:07:36.210 -- that's undefined. 00:07:36.210 --> 00:07:38.420 Let's do one more. 00:07:38.420 --> 00:07:41.890 What is -- I'm just going to think of random numbers -- 00:07:41.890 --> 00:07:44.930 4 divided by negative 1? 00:07:44.930 --> 00:07:50.610 Well, 4 divided by 1 is 4, but the signs are different. 00:07:50.610 --> 00:07:53.130 So it's negative 4. 00:07:53.130 --> 00:07:54.410 I hope that helps. 00:07:54.410 --> 00:07:57.680 Now what I want you to do is actually try as many of these 00:07:57.680 --> 00:08:01.380 multiplying and dividing negative numbers as you can. 00:08:01.380 --> 00:08:03.010 And you click on hints and it'll remind you 00:08:03.010 --> 00:08:04.260 of which rule to use. 00:08:07.460 --> 00:08:09.460 In your own time you might want to actually think about why 00:08:09.460 --> 00:08:13.490 these rules apply and what it means to multiply a negative 00:08:13.490 --> 00:08:15.320 number times a positive number. 00:08:15.320 --> 00:08:17.520 And even more interesting, what it means to multiply a negative 00:08:17.520 --> 00:08:20.230 number times a negative number. 00:08:20.230 --> 00:08:24.800 But I think at this point, hopefully, you are ready to 00:08:24.800 --> 00:08:27.160 start doing some problems. 00:08:27.160 --> 00:08:28.620 Good luck.

Adding/subtracting negative numbers

https://www.youtube.com/watch?v=C38B33ZywWs

vtt

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en

WEBVTT Kind: captions Language: en 00:00:01.060 --> 00:00:03.720 Welcome to the presentation on adding and subtracting 00:00:03.720 --> 00:00:04.440 negative numbers. 00:00:04.440 --> 00:00:06.390 So let's get started. 00:00:06.390 --> 00:00:08.530 So what is a negative number, first of all? 00:00:08.530 --> 00:00:12.090 Well, let me draw a number line. 00:00:12.090 --> 00:00:13.910 Well it's not much of a line but I think you'll 00:00:13.910 --> 00:00:15.200 get the picture. 00:00:15.200 --> 00:00:18.930 So we're used to the positive numbers, so if that's 0, you 00:00:18.930 --> 00:00:24.680 have 1, you have 2, you have 3, you have 4, and you keep going. 00:00:24.680 --> 00:00:28.640 And if I were to say what's 2 plus 2, you'd start at 2 and 00:00:28.640 --> 00:00:30.790 then you'd add 2 and you'd get to 4. 00:00:30.790 --> 00:00:32.160 I mean most of us it's second nature. 00:00:32.160 --> 00:00:33.940 But if you actually drew it on a number line 00:00:33.940 --> 00:00:36.140 you'd say 2 plus 2 is 4. 00:00:36.140 --> 00:00:38.490 And if I asked you what's 2 minus 1 or let's 00:00:38.490 --> 00:00:39.840 say what's 3 minus 2? 00:00:39.840 --> 00:00:44.280 If you start at 3 and you subtracted 2, you 00:00:44.280 --> 00:00:45.350 would end up at 1. 00:00:45.350 --> 00:00:51.310 That's 2 plus 2 is equal to 4 and 3 minus 2 is equal to 1. 00:00:51.310 --> 00:00:52.990 And this is a joke for you. 00:00:52.990 --> 00:00:56.990 Now what if I were to say what is 1 minus 3? 00:00:56.990 --> 00:00:58.160 Huh. 00:00:58.160 --> 00:01:00.110 Well, it's the same thing. 00:01:00.110 --> 00:01:04.370 You start at 1 and we're going to go 1 -- well, now we're 00:01:04.370 --> 00:01:06.970 going to go below 0 -- what happens below 0? 00:01:06.970 --> 00:01:08.510 Well then you start going to the negative numbers. 00:01:08.510 --> 00:01:14.940 Negative 1, negative 2, negative 3, and so on. 00:01:14.940 --> 00:01:22.550 So if I start at 1 right here, so 1 minus 3, so I go 1, 2, 00:01:22.550 --> 00:01:25.620 3, I end up at negative 2. 00:01:25.620 --> 00:01:30.690 So 1 minus 3 is equal to negative 2. 00:01:30.690 --> 00:01:32.290 This is something that you're probably already doing 00:01:32.290 --> 00:01:33.100 in your everyday life. 00:01:33.100 --> 00:01:36.870 If I were to tell you that boy, it's very cold today, it's 1 00:01:36.870 --> 00:01:40.830 degree, but tomorrow it's going to be 3 degrees colder, you 00:01:40.830 --> 00:01:42.710 might already know intuitively, well then we're going to be 00:01:42.710 --> 00:01:45.810 at a temperature of negative 2 degrees. 00:01:45.810 --> 00:01:48.050 So that's all a negative number means. 00:01:48.050 --> 00:01:51.870 And just remember when a negative number is big, so like 00:01:51.870 --> 00:01:58.100 negative 50, that's actually colder than negative 20, right? 00:01:58.100 --> 00:02:00.920 So a negative 50 is actually even a smaller number than 00:02:00.920 --> 00:02:03.890 negative 20 because it's even further to the left 00:02:03.890 --> 00:02:04.720 of negative 20. 00:02:04.720 --> 00:02:07.170 That's just something you'll get an intuitive feel for. 00:02:07.170 --> 00:02:09.400 Sometimes when you start you feel like oh, 50's a bigger 00:02:09.400 --> 00:02:11.770 number than 20, but it's a negative 50 as opposed 00:02:11.770 --> 00:02:13.860 to a positive 50. 00:02:13.860 --> 00:02:16.470 So let's do some problems, and I'm going to keep using the 00:02:16.470 --> 00:02:19.290 number line because I think it's useful. 00:02:19.290 --> 00:02:26.790 So let's do the problem 5 minus 12. 00:02:26.790 --> 00:02:28.810 I think you already might have an intuition of 00:02:28.810 --> 00:02:29.896 what this equals. 00:02:29.896 --> 00:02:34.600 But let me draw a line, 5 minus 12. 00:02:44.910 --> 00:02:52.936 So let me start with minus 10, minus 9, minus 8 --I think I'm 00:02:52.936 --> 00:02:58.170 going to run out of space -- minus 7, minus 6, minus 5 -- I 00:02:58.170 --> 00:03:03.380 should have this pre-drawn -- minus 4, minus 3, minus 2, 00:03:03.380 --> 00:03:11.470 minus 1, 0, 1, 2, 3, 4, I'll put 5 right here. 00:03:14.410 --> 00:03:15.570 5 minus 12. 00:03:15.570 --> 00:03:18.440 So if we start at 5 -- let me use a different color -- we 00:03:18.440 --> 00:03:20.945 start at 5 right here and we're going to go to the left 12 00:03:20.945 --> 00:03:22.800 because we're subtracting 12. 00:03:22.800 --> 00:03:33.880 So then we go 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 00:03:33.880 --> 00:03:34.780 Negative 7. 00:03:38.300 --> 00:03:40.010 That's pretty interesting. 00:03:40.010 --> 00:03:44.020 Because it also happens to be that 12 minus 5 00:03:44.020 --> 00:03:45.590 is equal to positive 7. 00:03:45.590 --> 00:03:48.910 So, I want you to think a little bit about why that is. 00:03:48.910 --> 00:03:53.980 Why the difference between 12 and 5 is 7, and the difference 00:03:53.980 --> 00:03:59.200 between -- well, I guess it's either way. 00:03:59.200 --> 00:04:01.025 Because in this situation we're also saying that the difference 00:04:01.025 --> 00:04:04.730 between 5 and 12 is negative 7, but the numbers are that far 00:04:04.730 --> 00:04:07.270 apart, but now we're starting with the lower number. 00:04:07.270 --> 00:04:09.640 I think that last sentence just completely confused you, but 00:04:09.640 --> 00:04:11.620 we'll keep moving forward. 00:04:11.620 --> 00:04:18.070 We just said 5 minus 12 is equal to minus 7. 00:04:18.070 --> 00:04:19.480 Let's do another one. 00:04:19.480 --> 00:04:25.720 What's negative 3 plus 5 equals what? 00:04:25.720 --> 00:04:28.060 Well, let's use the same number line. 00:04:28.060 --> 00:04:31.490 Let's go to negative 3 plus 5. 00:04:31.490 --> 00:04:33.590 So we're going to go to the right 5. 00:04:33.590 --> 00:04:38.530 1, 2, 3, 4, 5. 00:04:38.530 --> 00:04:38.950 It's a 2. 00:04:42.415 --> 00:04:44.230 It equals 2. 00:04:44.230 --> 00:04:48.790 So negative 3 plus 5 is equal to 2. 00:04:48.790 --> 00:04:53.720 That's interesting because 5 minus 3 is also equal to 2. 00:04:53.720 --> 00:04:57.340 Well, it turns out that 5 minus 3 is the same thing, it's just 00:04:57.340 --> 00:05:00.990 another way of writing 5 plus negative 3 or 00:05:00.990 --> 00:05:04.050 negative 3 plus 5. 00:05:04.050 --> 00:05:07.400 A general, easy way to always do negative numbers is it's 00:05:07.400 --> 00:05:11.650 just like regular subtraction and addition and subtraction, 00:05:11.650 --> 00:05:17.026 but now when we subtract we can go to the left below 0. 00:05:17.026 --> 00:05:19.720 Let's do another one. 00:05:19.720 --> 00:05:27.000 So what happens when you get let's say 2 minus minus 3? 00:05:27.000 --> 00:05:30.960 Well, if you think about how it should work out I think 00:05:30.960 --> 00:05:32.030 this will make sense. 00:05:32.030 --> 00:05:34.720 But it turns out that the negative number, the negative 00:05:34.720 --> 00:05:36.150 signs actually cancel out. 00:05:36.150 --> 00:05:40.610 So this is the same thing as 2 plus plus 3, and 00:05:40.610 --> 00:05:43.060 that just equals 5. 00:05:43.060 --> 00:05:46.530 Another way you could say is -- let's do another one -- what 00:05:46.530 --> 00:05:52.630 is negative 7 minus minus 2? 00:05:52.630 --> 00:05:58.030 Well that's the same thing as negative 7 plus 2. 00:05:58.030 --> 00:06:00.400 And remember, so we're doing to start at negative 7 and we're 00:06:00.400 --> 00:06:03.570 going to move two to the right. 00:06:03.570 --> 00:06:06.475 So if we move one to the right we go to negative 6, and then 00:06:06.475 --> 00:06:08.340 we move two to the right we get negative 5. 00:06:11.410 --> 00:06:13.950 That makes sense because negative 7 plus 2, that's the 00:06:13.950 --> 00:06:16.550 same thing as 2 minus 7. 00:06:16.550 --> 00:06:21.720 If it's 2 degrees and it gets 7 degrees colder, it's minus 5. 00:06:21.720 --> 00:06:23.240 Let's do a bunch of these. 00:06:23.240 --> 00:06:25.360 I think the more you do the more practice you have, and the 00:06:25.360 --> 00:06:29.000 modules explain it pretty well, probably better than I do. 00:06:29.000 --> 00:06:30.990 So let's just do a ton of problems. 00:06:30.990 --> 00:06:36.350 So if I said negative 7 minus 3. 00:06:36.350 --> 00:06:37.760 Well, now we're going to go three to the 00:06:37.760 --> 00:06:39.410 left of negative 7. 00:06:39.410 --> 00:06:41.740 We're going to get 3 less than negative 7 so that's 00:06:41.740 --> 00:06:45.430 negative 10, right? 00:06:45.430 --> 00:06:49.280 That makes sense, because if we had positive 7 plus 3 we're at 00:06:49.280 --> 00:06:51.880 7 to the right of 0 and we're going to go three more to the 00:06:51.880 --> 00:06:54.045 right of 0 and we get positive 10. 00:06:54.045 --> 00:06:57.580 So for 7 to the left of 0 and go three more to the left we're 00:06:57.580 --> 00:06:58.690 going to get negative 10. 00:06:58.690 --> 00:06:59.470 Let's do a bunch more. 00:06:59.470 --> 00:07:02.110 I know I'm probably confusing you, but practice is what's 00:07:02.110 --> 00:07:04.120 going to really help us. 00:07:04.120 --> 00:07:12.180 So say 3 minus minus 3, well, these negatives cancel out 00:07:12.180 --> 00:07:14.580 so that just equals 6. 00:07:14.580 --> 00:07:16.990 What's 3 minus 3? 00:07:16.990 --> 00:07:20.410 Well, that's easy that's just 0. 00:07:20.410 --> 00:07:23.990 What's minus 3 minus 3? 00:07:23.990 --> 00:07:26.420 Well now we're going to get three less than minus 3, 00:07:26.420 --> 00:07:28.890 well that's minus is 6. 00:07:28.890 --> 00:07:34.040 What's minus 3 minus minus 3? 00:07:34.040 --> 00:07:35.510 Interesting. 00:07:35.510 --> 00:07:40.800 Well, the minuses cancel out so you get minus 3 plus 3. 00:07:40.800 --> 00:07:43.530 Well, if we start three to the left of 0 and we move three to 00:07:43.530 --> 00:07:46.070 the right we end up at 0 again. 00:07:46.070 --> 00:07:48.260 So that makes sense, right? 00:07:48.260 --> 00:07:49.370 Let me do that again. 00:07:49.370 --> 00:07:53.290 Minus 3 minus minus 3. 00:07:53.290 --> 00:07:56.400 Anything minus itself should equal 0, right? 00:07:56.400 --> 00:07:57.960 That's why that equals 0. 00:07:57.960 --> 00:08:00.170 And that's why it makes sense that those two negatives 00:08:00.170 --> 00:08:02.160 cancel out and that's the same thing as this. 00:08:05.520 --> 00:08:07.430 Let's do a bunch more. 00:08:07.430 --> 00:08:12.150 Let's do 12 minus 13. 00:08:12.150 --> 00:08:13.620 That's pretty easy. 00:08:13.620 --> 00:08:17.950 Well, 12 minus 12 is 0, so 12 minus 13 is negative 1 00:08:17.950 --> 00:08:20.930 because we're going to go one the left of 0. 00:08:20.930 --> 00:08:24.670 Let's do 8 minus 5. 00:08:24.670 --> 00:08:27.280 Well, this one is just a normal problem, that's 3. 00:08:27.280 --> 00:08:29.650 What's 5 minus 8? 00:08:29.650 --> 00:08:33.080 Well, we're going to go all the way to 0 and then 3 more to the 00:08:33.080 --> 00:08:35.430 left of zero, so it's minus 3. 00:08:35.430 --> 00:08:38.510 I could draw a number line here. 00:08:38.510 --> 00:08:46.180 If this is 0 this is 5, and now we're going to go to left 8, 00:08:46.180 --> 00:08:48.310 then we end up and negative 3. 00:08:48.310 --> 00:08:49.410 You could do that for all of these. 00:08:49.410 --> 00:08:51.780 That actually might be a good exercise. 00:08:51.780 --> 00:08:54.590 I think this will give you good introduction and I recommend 00:08:54.590 --> 00:08:57.030 that you just do the modules because the modules actually, 00:08:57.030 --> 00:08:59.450 especially if you do the hints, it has a pretty nice graphic 00:08:59.450 --> 00:09:01.350 that's a lot nicer than anything I could draw 00:09:01.350 --> 00:09:02.870 on this chalkboard. 00:09:02.870 --> 00:09:05.740 So try that out and I'm going to try to record some more 00:09:05.740 --> 00:09:09.150 modules that hopefully won't confuse you as badly. 00:09:09.150 --> 00:09:11.210 You could also attend the seminar on adding and 00:09:11.210 --> 00:09:12.700 subtracting negative numbers. 00:09:12.700 --> 00:09:14.170 I hope you have fun. 00:09:14.170 --> 00:09:15.470 Bye.

systems of equations

https://www.youtube.com/watch?v=nok99JOhcjo

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https://www.youtube.com/api/timedtext?v=nok99JOhcjo&ei=gmeUZbTMM9CemLAPlICHqAY&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249842&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=86953ED0A463F8267A846450618D058AD570F482.E6EEBB18CC7C06EB6D18DA94C4DAE4B668C0F4E5&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.050 --> 00:00:04.040 Welcome to the presentation on systems of linear equations. 00:00:04.040 --> 00:00:06.970 So let's get started and see what it's all about. 00:00:06.970 --> 00:00:10.110 So let's say I had two equations now. 00:00:10.110 --> 00:00:15.740 The first equation let me write it as 9x minus 00:00:15.740 --> 00:00:21.760 4y equals minus 78. 00:00:21.760 --> 00:00:28.950 And the second equation I will write as 4x plus 00:00:28.950 --> 00:00:33.390 y is equal to mine 18. 00:00:33.390 --> 00:00:35.411 Now what we're going to do now is we're actually going to 00:00:35.411 --> 00:00:39.700 use both equations to solve for x and y. 00:00:39.700 --> 00:00:41.900 We already know that if you have one equation, it has one 00:00:41.900 --> 00:00:44.280 variable, it is very easy to solve for that one variable. 00:00:44.280 --> 00:00:45.790 But now we have to equations. 00:00:45.790 --> 00:00:47.340 You can almost view them as two constraints. 00:00:47.340 --> 00:00:50.340 And we're going to solve for both variables. 00:00:50.340 --> 00:00:51.790 And you might be a little confused. 00:00:51.790 --> 00:00:52.520 How does that work? 00:00:52.520 --> 00:00:54.910 Is it just magic that two equations can solves 00:00:54.910 --> 00:00:55.900 for two variables? 00:00:55.900 --> 00:00:56.800 Well it's not. 00:00:56.800 --> 00:00:58.850 Because you can actually rearranged each of these 00:00:58.850 --> 00:01:01.840 equations so that they look kind of in normal y 00:01:01.840 --> 00:01:03.700 equals mx plus b format. 00:01:03.700 --> 00:01:06.200 And I'm not going to draw these actual two equations because I 00:01:06.200 --> 00:01:08.860 don't know what they look like, but if this was a coordinate 00:01:08.860 --> 00:01:11.620 axis-- and I don't know what that first line actually does 00:01:11.620 --> 00:01:14.010 look like, we could do another model where we figured it out 00:01:14.010 --> 00:01:16.500 --but lets just say for sake of argument, that first line all 00:01:16.500 --> 00:01:20.540 the x's and y's that satisfy 9x minus 4y equals negative 00:01:20.540 --> 00:01:22.690 78, let's say it looks something like that. 00:01:22.690 --> 00:01:26.400 And let's say all of the x's and y's that satisfy that 00:01:26.400 --> 00:01:31.340 second equation, 4x plus y equals negative 18, let's say 00:01:31.340 --> 00:01:34.680 that looks something like this. 00:01:34.680 --> 00:01:35.620 Right? 00:01:35.620 --> 00:01:40.050 So, on the line is all of the x's and y's that satisfy this 00:01:40.050 --> 00:01:42.555 equation, and on the green line are all the x's and y's 00:01:42.555 --> 00:01:44.275 that satisfy this equation. 00:01:44.275 --> 00:01:48.170 But there's only one pair of x and y's that satisfy both 00:01:48.170 --> 00:01:51.430 equations, and you can guess where that is, that's 00:01:51.430 --> 00:01:52.560 right here right. 00:01:52.560 --> 00:01:57.660 Whatever that point is-- I'll do it in pink for emphasis. 00:01:57.660 --> 00:02:00.800 Whatever this point is, notice it's on both lines. 00:02:00.800 --> 00:02:05.260 So whatever x and y that is would be the solution to 00:02:05.260 --> 00:02:06.670 this system of equations. 00:02:06.670 --> 00:02:09.860 So let's actually figure out how to do that. 00:02:09.860 --> 00:02:12.080 So what we want to do is eliminate a variable, because 00:02:12.080 --> 00:02:15.200 if you can eliminate a variable then we can just solve for 00:02:15.200 --> 00:02:16.430 the one that's left over. 00:02:16.430 --> 00:02:19.930 And the way to do that-- let's see, I want to eliminate, I 00:02:19.930 --> 00:02:22.210 feel like eliminating this y, and I think you'll get 00:02:22.210 --> 00:02:24.630 an intuition for how we can do that later on. 00:02:24.630 --> 00:02:26.620 And the way I'm going to do that is I'm going to make 00:02:26.620 --> 00:02:29.250 it so that when I had this to this, they cancel out. 00:02:29.250 --> 00:02:31.340 Well, they don't cancel out right now, so I have to 00:02:31.340 --> 00:02:34.380 multiply this bottom equation by 4, and I think it'll be 00:02:34.380 --> 00:02:35.520 obvious why I'm doing it. 00:02:35.520 --> 00:02:37.810 So let's multiply this bottom equation by 4. 00:02:37.810 --> 00:02:50.820 And I get 16x plus 4y is equal to 40 plus 32 minus 72. 00:02:50.820 --> 00:02:51.130 Right? 00:02:51.130 --> 00:02:53.950 All I did is I multiplied both sides of the 00:02:53.950 --> 00:02:55.620 equation by 4, right? 00:02:55.620 --> 00:02:57.210 And you have to multiply every term because 00:02:57.210 --> 00:02:59.500 it's the distributive property on both sides. 00:02:59.500 --> 00:03:01.050 Whatever you do to one side you have to do to the other. 00:03:01.050 --> 00:03:03.300 Let me rewrite top equation again. 00:03:03.300 --> 00:03:05.230 And I'll write in the same color so we can keep 00:03:05.230 --> 00:03:06.340 track of things. 00:03:06.340 --> 00:03:13.360 9x minus 4y is equal to minus 78. 00:03:13.360 --> 00:03:18.580 OK, well now, if we were to add these two equations, when you 00:03:18.580 --> 00:03:20.430 add equations, you just add the left side and you 00:03:20.430 --> 00:03:22.270 add the right side. 00:03:22.270 --> 00:03:25.440 Well when you add, you have 16x plus 9x. 00:03:25.440 --> 00:03:28.590 Well that equals 25x. 00:03:28.590 --> 00:03:28.950 Right? 00:03:28.950 --> 00:03:31.450 16 plus 9. 00:03:31.450 --> 00:03:34.910 4y minus 4, that just equals 0. 00:03:34.910 --> 00:03:43.680 So that's plus 0 equals, and then we have minus 72 minus 78. 00:03:43.680 --> 00:03:51.490 So, let's see that's minus 150, minus 150, right? 00:03:51.490 --> 00:03:53.060 Just adding them all together. 00:03:53.060 --> 00:03:58.820 So we have 25x equals 150. 00:03:58.820 --> 00:04:03.420 Well, we could just divide both sides by 25 or multiply both 00:04:03.420 --> 00:04:05.380 sides by 1/25, it's the same thing. 00:04:05.380 --> 00:04:08.470 And you get x equals-- that's a negative 150 00:04:08.470 --> 00:04:11.500 --x equals minus 6. 00:04:11.500 --> 00:04:14.870 There we solved the x-coordinate. 00:04:14.870 --> 00:04:16.950 Now to solve the y-coordinate we can just use either one of 00:04:16.950 --> 00:04:18.500 these equations up at top. 00:04:18.500 --> 00:04:20.810 So let's use this one, it seems a little bit, 00:04:20.810 --> 00:04:23.020 marginally simpler. 00:04:23.020 --> 00:04:26.090 So we just substitute the x back in there and we get 00:04:26.090 --> 00:04:34.716 4 time minus 6 plus y is equal to minus 18. 00:04:34.716 --> 00:04:35.730 Go up here. 00:04:35.730 --> 00:04:42.565 4 times minus 6 we get minus 24 plus y is equal to minus 18. 00:04:42.565 --> 00:04:47.406 And then get y is equal to 24 minus 18. 00:04:47.406 --> 00:04:50.510 So y is equal to 6. 00:04:50.510 --> 00:04:54.100 So these two lines or these two equations, you could even say, 00:04:54.100 --> 00:05:00.300 intersect at the point x is m inus six and y is plus 6. 00:05:00.300 --> 00:05:02.520 So they actually intersect someplace around here instead. 00:05:02.520 --> 00:05:05.640 I drew these, the line probably look something more like that. 00:05:05.640 --> 00:05:06.950 But that's pretty cool, no? 00:05:06.950 --> 00:05:11.830 We actually solved for two variables using two equations. 00:05:11.830 --> 00:05:12.640 Let's see how much time I have. 00:05:12.640 --> 00:05:14.470 I think we have enough time to do another problem. 00:05:20.200 --> 00:05:23.020 So let's say I had the points-- and I'm going to write them in 00:05:23.020 --> 00:05:32.940 two different colors again --minus 7x minus 4y equals 9, 00:05:32.940 --> 00:05:39.150 and then the second equation is going to be x plus 00:05:39.150 --> 00:05:42.460 2y is equal to 3. 00:05:42.460 --> 00:05:45.140 Now if I were doing this as fast as possible, I'd probably 00:05:45.140 --> 00:05:47.990 multiply this equation times 7 and it would automatically 00:05:47.990 --> 00:05:49.020 cancel out. 00:05:49.020 --> 00:05:49.850 But that's easy way. 00:05:49.850 --> 00:05:51.290 I'm going to show you that sometimes you might have to 00:05:51.290 --> 00:05:54.780 multiply both equations-- actually, not in this case. 00:05:54.780 --> 00:05:56.800 Actually let's just do it the fast way real fast. 00:05:56.800 --> 00:05:59.380 So let's multiply this bottom equation by 7. 00:05:59.380 --> 00:06:00.830 And the whole reason why I want to the, multiply it with 7, 00:06:00.830 --> 00:06:03.440 because I want this to cancel out with this. 00:06:03.440 --> 00:06:10.150 If you multiply it by 7 you get 7x plus 14y is equal to 21. 00:06:10.150 --> 00:06:12.930 Let's write that first equation down again. 00:06:12.930 --> 00:06:19.065 Minus 7x minus 4y is equal to 9. 00:06:19.065 --> 00:06:20.330 Now we just add. 00:06:20.330 --> 00:06:24.260 This is a positive 7x, it just always looks like a negative. 00:06:24.260 --> 00:06:25.900 OK, so that's 0. 00:06:25.900 --> 00:06:32.460 14 minus 4y plus 10y is equal to 30. 00:06:32.460 --> 00:06:34.750 y is equal to 3. 00:06:34.750 --> 00:06:36.350 Now we just substitute back into either equation, 00:06:36.350 --> 00:06:37.980 lets do that one. 00:06:37.980 --> 00:06:42.110 x plus 2 times y, 2 times 3. 00:06:42.110 --> 00:06:43.880 x plus 6 equals 3. 00:06:43.880 --> 00:06:45.900 We get x equals negative 3. 00:06:45.900 --> 00:06:48.470 That one was super easy. 00:06:48.470 --> 00:06:49.550 The intercept. 00:06:49.550 --> 00:06:51.210 Hope I didn't do it to fast. 00:06:51.210 --> 00:06:54.430 Well, you can pause it and watch it again if you have. 00:06:54.430 --> 00:07:00.270 OK, so these two lines intersect at the point 00:07:00.270 --> 00:07:03.182 negative 3 comma 3. 00:07:03.182 --> 00:07:04.250 Let's do one more. 00:07:07.456 --> 00:07:10.710 Hope this one's harder. 00:07:10.710 --> 00:07:11.510 I think it will. 00:07:11.510 --> 00:07:20.300 OK, negative 3x minus 9y is equal to 66. 00:07:20.300 --> 00:07:27.200 We have minus 7x plus 4y is equal to minus 71. 00:07:27.200 --> 00:07:28.370 So here it's not obvious. 00:07:28.370 --> 00:07:31.540 What we have to do is, let's say we want to cancel 00:07:31.540 --> 00:07:33.980 out the y's first. 00:07:33.980 --> 00:07:36.500 What we do is we try to make both of them equal to the least 00:07:36.500 --> 00:07:38.660 common multiple of 9 and 4. 00:07:38.660 --> 00:07:43.340 So, if we multiply the top equation by 4 we get-- 00:07:43.340 --> 00:07:44.520 I'll do it right here. 00:07:44.520 --> 00:07:45.870 Let's multiply it by 4. 00:07:45.870 --> 00:07:47.960 Times 4. 00:07:47.960 --> 00:07:59.200 We'll get minus 12x minus 36y is equal to 4 times 00:07:59.200 --> 00:08:05.400 240 plus 24 is 264. 00:08:05.400 --> 00:08:06.930 Right, I hope that's right. 00:08:06.930 --> 00:08:09.220 We multiply the second equation by 9. 00:08:09.220 --> 00:08:25.420 So it's minus 63x plus 36y is equal to, let's see, 639. 00:08:25.420 --> 00:08:26.030 Big numbers. 00:08:26.030 --> 00:08:29.350 639. 00:08:29.350 --> 00:08:31.540 OK, now we add the two equations. 00:08:31.540 --> 00:08:43.570 Minus 12 minus 63 thats minus 75x-- these cancel out --equals 00:08:43.570 --> 00:08:50.130 264, let's see what's 639 minus 264. 00:08:50.130 --> 00:08:51.160 See I do this in real time. 00:08:51.160 --> 00:08:55.100 I don't use some kind of solution manual or something. 00:08:55.100 --> 00:08:59.710 13 and 5, 70. 00:08:59.710 --> 00:09:02.260 I don't know if I'm right, but we'll see. 00:09:02.260 --> 00:09:06.360 Since it's actually the negative 639, this is minus 00:09:06.360 --> 00:09:12.440 375, and I know that seventy five goes into 300 4 00:09:12.440 --> 00:09:16.450 times, so x is equal to 5. 00:09:16.450 --> 00:09:19.515 75 times 5 is 375. 00:09:19.515 --> 00:09:22.460 We just divided both sides by 75. 00:09:22.460 --> 00:09:25.367 So if x is 5 we just substitute it back into-- let's 00:09:25.367 --> 00:09:27.890 use this equation. 00:09:27.890 --> 00:09:36.380 So we get minus 3 times 5 minus 9y is equal to 66. 00:09:36.380 --> 00:09:41.920 We get minus 15 minus 9y equals 66. 00:09:41.920 --> 00:09:45.880 Minus 9y is equal to 81. 00:09:45.880 --> 00:09:49.840 And then we get y is equal to minus 9. 00:09:49.840 --> 00:09:53.530 So the answer is 5 comma minus 9. 00:09:53.530 --> 00:09:55.530 I think you're ready to do some systems of equations now. 00:09:55.530 --> 00:09:57.090 Have Fun.

Algebra: Slope 3

https://www.youtube.com/watch?v=8XffLj2zvf4

vtt

https://www.youtube.com/api/timedtext?v=8XffLj2zvf4&ei=gmeUZbTIO7WEp-oPgrSL8Aw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=90C5E27D12E6F1228DB3A73DE998311DB42D7991.1B94A0ABF2B2BB45FD2E129EDE14C20C7E6F2AB2&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:00.940 --> 00:00:02.610 Well, I had to stop that last presentation because 00:00:02.610 --> 00:00:03.630 I ran out of time. 00:00:03.630 --> 00:00:07.010 But now we can start up right where we left off. 00:00:07.010 --> 00:00:09.490 So far, we had figured out the slope and it made sense to 00:00:09.490 --> 00:00:12.350 us because we figured out the slope was minus 4/3. 00:00:15.390 --> 00:00:16.250 You could look at it two ways. 00:00:16.250 --> 00:00:18.960 When the run is 3, the rise is negative 4. 00:00:18.960 --> 00:00:22.730 So rise we went down by negative 4, we ran 3, and 00:00:22.730 --> 00:00:25.310 you can keep doing that on the line and you'll keep 00:00:25.310 --> 00:00:26.340 ending up on the line. 00:00:26.340 --> 00:00:30.330 You could've also gone down 8 and then moved over 6. 00:00:30.330 --> 00:00:32.250 Or you could move over 6 and go down 8. 00:00:32.250 --> 00:00:34.640 If you went down 8 you would've showed up here. 00:00:34.640 --> 00:00:36.660 Then you would've come down back here. 00:00:36.660 --> 00:00:40.880 Whatever the slope is, if you change the y by the rise, and 00:00:40.880 --> 00:00:43.010 you change the run by the x, you should end up 00:00:43.010 --> 00:00:44.360 on the line again. 00:00:44.360 --> 00:00:47.420 I know I'm doing this over and over again, but I think 00:00:47.420 --> 00:00:48.730 it'll slowly sink in. 00:00:48.730 --> 00:00:52.030 But anyway, where we left off we were going to solve for b so 00:00:52.030 --> 00:00:55.090 that we could figure out the final equation for this line. 00:00:55.090 --> 00:00:57.780 Well, just like we did the last time, we just have to 00:00:57.780 --> 00:01:01.240 substitute a y or an x that we know works for this line. 00:01:01.240 --> 00:01:03.100 We know either of these coordinates will work, 00:01:03.100 --> 00:01:04.600 and then we solve for b. 00:01:04.600 --> 00:01:07.500 Let's try this first one: 2 comma negative 3. 00:01:07.500 --> 00:01:10.585 So y is negative 3. 00:01:10.585 --> 00:01:13.070 Don't get confused between x and y. 00:01:13.070 --> 00:01:19.980 y is negative 3 equals minus 4/3 times x, which is 2. 00:01:19.980 --> 00:01:21.570 Plus b. 00:01:21.570 --> 00:01:29.600 And then we get negative 3 equals negative 8/3 plus b. 00:01:29.600 --> 00:01:35.950 And then we get b is equal to-- well, 3 is equal to minus 9/3. 00:01:35.950 --> 00:01:38.130 And then I'm just going to put this on the other side. 00:01:38.130 --> 00:01:41.040 So that's plus 8/3. 00:01:41.040 --> 00:01:45.270 So we get minus 1/3 is the y-intercept. 00:01:45.270 --> 00:01:51.740 So the equation of this line is y is equal to minus 4/3 x and 00:01:51.740 --> 00:01:54.430 then this y-intercept is minus 1/3. 00:01:57.420 --> 00:02:00.110 I know this is extremely messy, but the equation of this line 00:02:00.110 --> 00:02:04.150 once again, is y equals minus 4/3 x minus 1/3. 00:02:04.150 --> 00:02:07.920 Now let's look at the graph and see if that makes sense. 00:02:07.920 --> 00:02:08.870 Well, let's see. 00:02:08.870 --> 00:02:10.810 The y-intercept is right here. 00:02:10.810 --> 00:02:13.070 That's where you intersect the y-axis at the point. 00:02:13.070 --> 00:02:15.320 And we would know exactly what that point is. 00:02:15.320 --> 00:02:19.700 It's 0 comma negative 1/3. 00:02:19.700 --> 00:02:21.585 That's the y-intercept, and it makes sense. 00:02:21.585 --> 00:02:23.770 And even when you look at an equation it's pretty obvious 00:02:23.770 --> 00:02:25.780 that this is going to be the y-intercept because when x 00:02:25.780 --> 00:02:27.530 equals 0 this term gets crossed out. 00:02:27.530 --> 00:02:30.090 Because 0 times negative 4/3. 00:02:30.090 --> 00:02:32.740 And then y would equal negative 1/3. 00:02:32.740 --> 00:02:37.130 Let's do one more just to bore you. 00:02:37.130 --> 00:02:40.790 And because my wife is a resident and she works 30 00:02:40.790 --> 00:02:43.150 hours of time and I have nothing better to do. 00:02:43.150 --> 00:02:44.280 All right. 00:02:44.280 --> 00:02:47.060 Let me put that graph back there. 00:02:47.060 --> 00:02:49.000 I joke, but you don't realize that it's true. 00:02:52.140 --> 00:02:53.810 Let's put the graph back there and I'm going to do 00:02:53.810 --> 00:02:55.770 another random points. 00:02:55.770 --> 00:02:57.570 I'm going to go a little faster this time. 00:02:57.570 --> 00:03:05.560 So let's say I had negative 8 comma 5 and I had-- let me 00:03:05.560 --> 00:03:11.840 think of a good one-- 2 comma-- I'm just going 00:03:11.840 --> 00:03:14.140 to make up a number. 00:03:14.140 --> 00:03:19.790 2 comma, let's say 0. 00:03:19.790 --> 00:03:20.800 That's interesting. 00:03:20.800 --> 00:03:21.560 2 comma 0. 00:03:21.560 --> 00:03:23.440 So let's graph negative 8 comma 5. 00:03:23.440 --> 00:03:27.670 1, 2, 3, 4, 5, 6, 7, 8. 00:03:27.670 --> 00:03:31.070 1, 2, 3, 4, 5. 00:03:31.070 --> 00:03:35.240 Right there is minus 8 comma 5. 00:03:35.240 --> 00:03:39.270 And then 1, 2, and then 0 is right here. 00:03:39.270 --> 00:03:41.940 So that's 2 comma 0. 00:03:41.940 --> 00:03:43.415 And now let me draw the line. 00:03:47.180 --> 00:03:47.740 Oh my God. 00:03:47.740 --> 00:03:48.770 I thought I was using the line tool. 00:03:48.770 --> 00:03:50.350 That's horrendous. 00:03:50.350 --> 00:03:51.210 I wish there was an undo tool. 00:03:53.940 --> 00:03:55.160 That was horrendous. 00:03:55.160 --> 00:03:57.300 That was unacceptable. 00:03:57.300 --> 00:03:59.452 That's a little bit better. 00:03:59.452 --> 00:04:01.460 Just so you know that the line doesn't end there. 00:04:01.460 --> 00:04:05.190 Lines go on forever and ever in the coordinate axis. 00:04:05.190 --> 00:04:06.480 There you go. 00:04:06.480 --> 00:04:07.390 OK. 00:04:07.390 --> 00:04:09.770 So let's figure out the slope of this line. 00:04:09.770 --> 00:04:13.420 Well, change in y over change in x. 00:04:13.420 --> 00:04:14.785 I'm using the line tool again. 00:04:14.785 --> 00:04:15.995 I'm a spaz today. 00:04:19.090 --> 00:04:20.940 Let's take this as a starting point. 00:04:20.940 --> 00:04:23.140 We'll get 5 minus 0. 00:04:23.140 --> 00:04:25.430 y sub 1 minus y sub 2. 00:04:25.430 --> 00:04:29.770 Over x sub 1-- minus 8 minus 2. 00:04:29.770 --> 00:04:33.650 So it equals 5 over minus 10. 00:04:33.650 --> 00:04:35.730 And that equals minus 1/2. 00:04:35.730 --> 00:04:39.372 So that means for every 2 we go over, we go down 1. 00:04:39.372 --> 00:04:43.150 Well, for every 1 we go down, we go over 2, 00:04:43.150 --> 00:04:43.800 which makes sense. 00:04:43.800 --> 00:04:47.410 If we go down 2, we'll go over 4. 00:04:47.410 --> 00:04:49.743 Because 2/4 is the same thing as 1/2. 00:04:49.743 --> 00:04:51.330 I hope that makes sense to you. 00:04:51.330 --> 00:04:53.630 I know this is a downwards sloping line. 00:04:53.630 --> 00:04:54.600 So that was fast. 00:04:54.600 --> 00:04:57.230 So we know that the equation of the line so far is y is 00:04:57.230 --> 00:05:02.170 equal to minus 1/2 x plus b. 00:05:02.170 --> 00:05:03.520 Now we just solve for b. 00:05:03.520 --> 00:05:06.100 Let's substitute some numbers in here. 00:05:06.100 --> 00:05:06.840 Well, let's use this one. 00:05:06.840 --> 00:05:08.610 This is interesting: 2 comma 0. 00:05:08.610 --> 00:05:10.620 So y is 0. 00:05:10.620 --> 00:05:15.280 Equals minus 1/2 times 2 plus b. 00:05:15.280 --> 00:05:19.130 Well, 0 is equal to and this is minus 1 plus b. 00:05:19.130 --> 00:05:21.180 And we get b equals 1. 00:05:21.180 --> 00:05:23.050 This is a pretty easy problem. 00:05:23.050 --> 00:05:29.320 So now we get y is equal to minus 1/2 x plus 1. 00:05:32.150 --> 00:05:35.110 OK, now let's see if this actually looks right 00:05:35.110 --> 00:05:35.890 on our problem. 00:05:35.890 --> 00:05:37.210 Well, this is telling us that the y-intercept 00:05:37.210 --> 00:05:39.310 is at the point 0, 1. 00:05:39.310 --> 00:05:41.030 0, 1 is right here. 00:05:41.030 --> 00:05:44.960 And our algebra confirms our drawing. 00:05:44.960 --> 00:05:47.320 This is the y-intercept and we see that the 00:05:47.320 --> 00:05:48.990 slope is negative 1/2. 00:05:48.990 --> 00:05:51.590 It makes sense because it's a downward sloping line, 00:05:51.590 --> 00:05:52.726 but it's not too steep. 00:05:55.350 --> 00:05:58.340 For every 1 that we go down, we go over 2. 00:05:58.340 --> 00:06:00.200 So that's negative 1/2. 00:06:00.200 --> 00:06:05.580 Or you could say for every 2 we run, we rise negative 1. 00:06:05.580 --> 00:06:07.810 Either one, we end up on a line again. 00:06:07.810 --> 00:06:09.280 So I hope that helps. 00:06:09.280 --> 00:06:13.130 I think you are definitely ready to do a lot of these 00:06:13.130 --> 00:06:15.450 slope problems that we have on the Khan Academy. 00:06:15.450 --> 00:06:17.690 So I hope you have fun and if you any questions you can 00:06:17.690 --> 00:06:20.410 attend one of the seminars. 00:06:20.410 --> 00:06:22.330 Have fun.

Algebra: Slope 2

https://www.youtube.com/watch?v=Kk9IDameJXk

vtt

https://www.youtube.com/api/timedtext?v=Kk9IDameJXk&ei=g2eUZb_iCcG_mLAPory-uAk&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=D12EFCE584C4E818D3DF9DE52A7ECF5216B2D474.0C7733C1AC4A7D507B7149856731B66C13A22FFE&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.650 --> 00:00:02.030 Hello. 00:00:02.030 --> 00:00:04.490 We're now going to do some more slope, and then maybe some 00:00:04.490 --> 00:00:06.550 y-intercept problems as well. 00:00:06.550 --> 00:00:08.660 Let's get started. 00:00:08.660 --> 00:00:11.090 So let me make up a problem. 00:00:11.090 --> 00:00:17.330 Let's say we have the points 2, 5. 00:00:17.330 --> 00:00:24.790 The other point, let's make that negative 3, negative 3. 00:00:24.790 --> 00:00:27.410 Well, first let's just graph those two points. 00:00:27.410 --> 00:00:29.760 I'm going to graph them in yellow. 00:00:29.760 --> 00:00:31.280 So 2, 5. 00:00:31.280 --> 00:00:33.920 Let's see that's one two. 00:00:33.920 --> 00:00:36.720 One, two, three, four, five. 00:00:36.720 --> 00:00:40.320 So 2, 5 is going to be right over there. 00:00:47.300 --> 00:00:48.310 OK. 00:00:48.310 --> 00:00:52.080 And then let me graph negative 3, negative 3. 00:00:52.080 --> 00:00:54.520 So it's one, two, three. 00:00:54.520 --> 00:00:56.260 One, two, three. 00:00:56.260 --> 00:00:59.820 So negative 3, negative 3 is right over there. 00:00:59.820 --> 00:01:03.750 And then now let me draw a line that will connect them. 00:01:10.920 --> 00:01:11.760 That's my new technique. 00:01:11.760 --> 00:01:14.720 I draw it in two pieces. 00:01:14.720 --> 00:01:15.800 I think that's good enough. 00:01:15.800 --> 00:01:16.170 OK. 00:01:16.170 --> 00:01:18.963 So let's see if we can at least first figure out the slope of 00:01:18.963 --> 00:01:20.640 the line, and then if we have time we'll try to figure 00:01:20.640 --> 00:01:21.470 out the y-intercept. 00:01:21.470 --> 00:01:23.990 And then we'll know the whole equation for the line. 00:01:23.990 --> 00:01:28.570 Let me pick a slightly thinner color, and we'll get started. 00:01:28.570 --> 00:01:32.060 So the slope, if you saw the last module that just 00:01:32.060 --> 00:01:34.040 introduces how we calculate the slope, that's 00:01:34.040 --> 00:01:35.430 just rise over run. 00:01:35.430 --> 00:01:41.730 Or, change in y over change in x. 00:01:41.730 --> 00:01:44.840 This is y. 00:01:44.840 --> 00:01:46.630 So let's just do that real fast. 00:01:46.630 --> 00:01:48.490 So let's take this as our starting point. 00:01:48.490 --> 00:01:51.430 So change in y could be 5-- remember, y is the 00:01:51.430 --> 00:01:57.930 second coordinate-- 5 minus negative 3. 00:01:57.930 --> 00:01:59.860 And that's this one. 00:01:59.860 --> 00:02:05.210 Over-- now you do the change in x-- 2 minus, this 00:02:05.210 --> 00:02:07.690 is also negative 3. 00:02:07.690 --> 00:02:11.360 Well 5 minus negative 3, that's 5 plus plus 3. 00:02:11.360 --> 00:02:13.190 So that equals 8. 00:02:13.190 --> 00:02:15.300 And then 2 minus negative 3. 00:02:15.300 --> 00:02:19.310 Once again that's 2 plus plus 3, so that equals 5. 00:02:19.310 --> 00:02:21.530 So we figured out the slope of this equation. 00:02:21.530 --> 00:02:23.060 It's 8/5. 00:02:23.060 --> 00:02:24.640 And let's see if that makes sense. 00:02:24.640 --> 00:02:27.280 Let's figure out what the rise and the run is. 00:02:27.280 --> 00:02:31.160 If we were to start at this point right here, let's see how 00:02:31.160 --> 00:02:34.750 much we have to rise to get to the same y-coordinate 00:02:34.750 --> 00:02:35.940 as the other point. 00:02:35.940 --> 00:02:37.400 So let's see. 00:02:37.400 --> 00:02:41.480 We're here, and the other point is up here. 00:02:41.480 --> 00:02:47.840 So let's figure out what this distance is. 00:02:47.840 --> 00:02:51.830 Actually, now is a good time to use the fat. 00:02:51.830 --> 00:02:54.110 Oh man, I have a shaky hand. 00:02:54.110 --> 00:02:54.530 OK. 00:02:54.530 --> 00:02:55.780 Let's figure out what that distance is. 00:02:55.780 --> 00:02:59.620 That distance is delta y, which is change in y. 00:02:59.620 --> 00:03:05.420 So it's one, two, three, four, five, six, seven, eight. 00:03:05.420 --> 00:03:07.170 That equals 8. 00:03:07.170 --> 00:03:08.710 And that makes sense, because if you think about it 00:03:08.710 --> 00:03:09.800 what did we just do? 00:03:09.800 --> 00:03:15.130 We just took y equals 5, which was up here, minus 00:03:15.130 --> 00:03:17.220 y equals negative 3. 00:03:17.220 --> 00:03:19.440 And so obviously we just calculated that distance 00:03:19.440 --> 00:03:23.240 just by looking at the two coordinates 5 minus negative 3. 00:03:23.240 --> 00:03:25.930 When you do this calculation it actually gives you 00:03:25.930 --> 00:03:27.590 this distance right here. 00:03:27.590 --> 00:03:29.990 So that's how we figure out how much we have to rise. 00:03:29.990 --> 00:03:32.550 So now let's do the run. 00:03:32.550 --> 00:03:35.170 Well the run, to go from this point to the other 00:03:35.170 --> 00:03:37.290 point, we went this far. 00:03:40.950 --> 00:03:43.300 And let's count how far that is. 00:03:43.300 --> 00:03:47.880 Well, it's one, two, three, four, five units. 00:03:47.880 --> 00:03:52.126 So we can say delta x is equal to 5. 00:03:52.126 --> 00:03:55.120 And that's exactly what we did. delta y over delta x was equal 00:03:55.120 --> 00:03:58.820 to 8/5, or rise over run is equal to 8/5. 00:03:58.820 --> 00:04:01.540 And it would have been the same thing if we calculated run here 00:04:01.540 --> 00:04:03.300 or if we calculated rise here. 00:04:03.300 --> 00:04:05.430 But it's the same thing. 00:04:05.430 --> 00:04:07.650 Hope that's making sense to you. 00:04:07.650 --> 00:04:11.020 And I hope that also makes sense that if the rise for a 00:04:11.020 --> 00:04:13.765 given run becomes more, then the slope of the line is going 00:04:13.765 --> 00:04:16.650 to become steeper and it'll become a bigger number. 00:04:16.650 --> 00:04:18.380 So let's see what we have so far for the 00:04:18.380 --> 00:04:19.490 equation of this line. 00:04:19.490 --> 00:04:23.680 So so far we know the equation of this line is equal to, y is 00:04:23.680 --> 00:04:31.710 equal to the slope 8/5 x plus b. 00:04:31.710 --> 00:04:32.740 So we're almost done. 00:04:32.740 --> 00:04:39.390 We just have to figure out this b right here. 00:04:39.390 --> 00:04:41.200 Now that b, just so you remember, that's 00:04:41.200 --> 00:04:42.400 the y-intercept. 00:04:42.400 --> 00:04:45.130 And that's where we intersect the y-axis. 00:04:45.130 --> 00:04:48.200 And since this graph is pretty neat, we can actually inspect 00:04:48.200 --> 00:04:50.530 it and see that, well, it looks like we're intersecting 00:04:50.530 --> 00:04:51.450 the y-axis at 2. 00:04:51.450 --> 00:04:54.460 So my guess is we're going to come up with b equals 2. 00:04:54.460 --> 00:04:56.170 But let's solve it, just in case we didn't have this 00:04:56.170 --> 00:04:58.020 neatly drawn graph here. 00:04:58.020 --> 00:05:00.200 So how can we solve for b? 00:05:00.200 --> 00:05:02.860 Well, we can substitute values that we know 00:05:02.860 --> 00:05:04.310 that work for x and y. 00:05:04.310 --> 00:05:07.280 Well either of these points are on that line, so we can 00:05:07.280 --> 00:05:09.340 substitute them in for x and y. 00:05:09.340 --> 00:05:12.310 So let's use the first one. 00:05:12.310 --> 00:05:12.730 OK. 00:05:12.730 --> 00:05:21.915 So the y we get 5, will equal 8/5 times x. 00:05:21.915 --> 00:05:24.050 Well, x there is 2. 00:05:24.050 --> 00:05:27.670 Times 2 plus b. 00:05:27.670 --> 00:05:37.080 Well, now we just get 5 is equal to 16/5 plus b. 00:05:37.080 --> 00:05:44.590 And then we get b equals-- well 5 is 25/5, right? 00:05:44.590 --> 00:05:53.080 5 is 25/5 minus 16/5 equals 9/5. 00:05:53.080 --> 00:05:53.320 All right. 00:05:53.320 --> 00:05:55.140 See, so I was actually wrong. 00:05:55.140 --> 00:05:58.150 When I looked at this graph I said, oh that looks like 00:05:58.150 --> 00:06:00.930 almost 2, so yeah it's probably going to be 2. 00:06:00.930 --> 00:06:03.780 But when we actually did it using algebra, when we did it 00:06:03.780 --> 00:06:07.010 analytically, we actually saw that b is equal to 9/5. 00:06:07.010 --> 00:06:08.340 So it's almost 2. 00:06:08.340 --> 00:06:11.480 9/5 is 1 and 4/5, or 1.8. 00:06:11.480 --> 00:06:13.430 So that's almost 2, but it actually turns 00:06:13.430 --> 00:06:14.240 out that it's not. 00:06:14.240 --> 00:06:15.610 It's at 1.8. 00:06:15.610 --> 00:06:16.870 And I can write it down as a decimal. 00:06:16.870 --> 00:06:18.290 1.8. 00:06:18.290 --> 00:06:20.330 So the final equation for the line, I'm going to try to 00:06:20.330 --> 00:06:26.210 squeeze it in at the bottom of this page, it's going to be y 00:06:26.210 --> 00:06:29.290 is equal to-- well, we know the slope. 00:06:29.290 --> 00:06:33.560 8/5 x. 00:06:33.560 --> 00:06:36.010 Now we just add the y-intercept. 00:06:36.010 --> 00:06:38.860 Plus 9/5. 00:06:38.860 --> 00:06:39.260 There. 00:06:39.260 --> 00:06:40.740 We solved it. 00:06:40.740 --> 00:06:41.390 Let's do another one. 00:06:41.390 --> 00:06:43.200 And so-- that's 9/5. 00:06:43.200 --> 00:06:43.840 I don't want to be too repetitive. 00:06:43.840 --> 00:06:46.400 Let's do another problem. 00:06:46.400 --> 00:06:48.920 Time to do another problem, and let me put that 00:06:48.920 --> 00:06:50.270 graph back there again. 00:06:53.080 --> 00:06:53.950 There you go. 00:06:53.950 --> 00:06:54.420 All right. 00:06:54.420 --> 00:06:57.180 I'm going to think of two random numbers again. 00:06:57.180 --> 00:06:59.410 Let me try to do this fast, because YouTube puts a 00:06:59.410 --> 00:07:01.610 10 minute limit on me. 00:07:01.610 --> 00:07:07.890 So let's say I had the points 2, negative 3. 00:07:07.890 --> 00:07:13.890 And I had the point negative 4, 5. 00:07:13.890 --> 00:07:15.240 So 2, negative 3. 00:07:15.240 --> 00:07:19.270 Let's plot that sucker real fast. 00:07:19.270 --> 00:07:21.810 So x is 2, so it's here. 00:07:21.810 --> 00:07:22.690 And the negative 3. 00:07:22.690 --> 00:07:24.120 One, two, three. 00:07:24.120 --> 00:07:26.510 So 2, negative 3 is there. 00:07:26.510 --> 00:07:28.080 And negative 4, 5. 00:07:28.080 --> 00:07:31.150 So that's one, two, three, four. 00:07:31.150 --> 00:07:33.180 One, two, three, four, five. 00:07:33.180 --> 00:07:35.180 I have to count like this because this 00:07:35.180 --> 00:07:36.560 graph is unlabeled. 00:07:36.560 --> 00:07:38.720 But if we actually were to draw in the coordinates you would 00:07:38.720 --> 00:07:43.750 that see this is 5, and this is negative 4, and so on. 00:07:43.750 --> 00:07:46.460 And this is 2, and this is negative 3. 00:07:46.460 --> 00:07:50.770 And now let's just draw a line. 00:07:50.770 --> 00:07:52.875 Let's draw it right there with my shaky hand. 00:07:57.150 --> 00:07:57.470 OK. 00:07:57.470 --> 00:07:59.030 There you go. 00:07:59.030 --> 00:07:59.850 Good line. 00:07:59.850 --> 00:08:03.470 And another good line. 00:08:03.470 --> 00:08:03.960 All right. 00:08:03.960 --> 00:08:05.910 So first we need to figure out the slope. 00:08:05.910 --> 00:08:09.240 Well we could just do that doing the algebra. 00:08:09.240 --> 00:08:13.760 So its slope is just delta-- I'm still using the line tool 00:08:13.760 --> 00:08:18.270 again-- delta y over delta x. 00:08:18.270 --> 00:08:20.730 Change in y over change in x. 00:08:20.730 --> 00:08:22.620 Let's take this y as the first point now. 00:08:22.620 --> 00:08:28.460 So we'll say 5 minus this y, negative 3. 00:08:30.960 --> 00:08:33.930 Over-- now since we used the 5 first we have to use the 00:08:33.930 --> 00:08:35.390 negative 4 first as well. 00:08:35.390 --> 00:08:39.360 Negative 4 minus 2. 00:08:39.360 --> 00:08:42.990 Well 5 minus negative 3, that equals 8. 00:08:42.990 --> 00:08:47.305 And negative 4 minus 2, well that equals negative 6. 00:08:47.305 --> 00:08:52.080 And negative 8/6, well that equals-- they're 00:08:52.080 --> 00:08:53.210 both divisible by 2. 00:08:53.210 --> 00:08:55.070 So that equals minus 4/3. 00:08:57.690 --> 00:08:59.740 And let's see, does that make sense as the slope? 00:08:59.740 --> 00:09:03.310 Well, if we were to go down four from this point. 00:09:03.310 --> 00:09:06.660 So if the rise was negative 4-- one, two, three, four. 00:09:06.660 --> 00:09:09.630 So if we go down-- woops, I'm using white. 00:09:09.630 --> 00:09:13.060 So that's why you can't see it. 00:09:13.060 --> 00:09:18.940 We go down by four here, and then we go to the right 00:09:18.940 --> 00:09:20.350 three, positive 3. 00:09:20.350 --> 00:09:21.580 We still end up on the line. 00:09:21.580 --> 00:09:23.190 So it works. 00:09:23.190 --> 00:09:23.950 Looks good to me. 00:09:23.950 --> 00:09:27.600 Let's see if I can solve the y-intercept in 30 seconds. 00:09:27.600 --> 00:09:29.170 Otherwise, I'll start it on the next module. 00:09:29.170 --> 00:09:35.970 So we get y is equal to minus 4/3 x, plus b. 00:09:35.970 --> 00:09:38.160 And actually what we'll do is we'll leave off here, and I'm 00:09:38.160 --> 00:09:40.540 going to solve for b-- and you could try to do it on your 00:09:40.540 --> 00:09:43.990 own-- in the next installment of this presentation.

Algebra: Slope

https://www.youtube.com/watch?v=hXP1Gv9IMBo

vtt

https://www.youtube.com/api/timedtext?v=hXP1Gv9IMBo&ei=gmeUZYXJOaiMp-oPrfur0AI&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=7707F0A195349D5420E7BD75476859631775D0F2.1D6E842EC76E3D898FD9980769D25668C266A6EE&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.050 --> 00:00:04.020 Welcome to the presentation on figuring out the slope. 00:00:04.020 --> 00:00:05.320 Let's get started. 00:00:05.320 --> 00:00:07.780 So, let's say I have two points. 00:00:07.780 --> 00:00:10.270 And, as we learned in previous presentations, that all 00:00:10.270 --> 00:00:12.080 you need to define a line is two points. 00:00:12.080 --> 00:00:15.200 And I think if you think about that, that makes sense. 00:00:15.200 --> 00:00:16.200 Let's say we have two points. 00:00:16.200 --> 00:00:19.450 And let me write down the two points we're going to have. 00:00:19.450 --> 00:00:25.020 Let's say one point is, why isn't it writing. 00:00:25.020 --> 00:00:26.846 Sometimes this thing acts a little finicky. 00:00:26.846 --> 00:00:30.080 Oh, that's because I was trying to write in black. 00:00:30.080 --> 00:00:38.160 Let's say that one point is, negative 1, 3. 00:00:38.160 --> 00:00:38.660 So, let's see. 00:00:38.660 --> 00:00:40.500 Where do we graph that? 00:00:40.500 --> 00:00:42.470 So, this is 0, 0. 00:00:42.470 --> 00:00:47.490 We go negative 1, this is negative 1 here. 00:00:47.490 --> 00:00:48.970 And then we're going to go 3 up. 00:00:48.970 --> 00:00:50.530 1, 2, 3. 00:00:50.530 --> 00:00:52.560 Because this is 3 right here. 00:00:52.560 --> 00:00:56.580 So, negative 1, 3 is going to be right over there. 00:00:56.580 --> 00:00:58.310 OK, so that's the first point. 00:00:58.310 --> 00:01:01.210 The second point, I'm going to do it in a different color. 00:01:01.210 --> 00:01:06.890 The second point is 2, 1. 00:01:06.890 --> 00:01:08.290 Let's see where we would put that. 00:01:08.290 --> 00:01:11.080 We would count 1, 2. 00:01:11.080 --> 00:01:13.610 This is 2, 1. 00:01:13.610 --> 00:01:14.670 Because this is 1. 00:01:14.670 --> 00:01:17.530 So the point's going to be here. 00:01:17.530 --> 00:01:19.710 So we've graphed our two points. 00:01:19.710 --> 00:01:23.600 And now the line that connects them, it's going to look 00:01:23.600 --> 00:01:25.450 something thing like this. 00:01:25.450 --> 00:01:27.060 And I hope I can draw it well. 00:01:36.300 --> 00:01:39.078 Through that point. 00:01:39.078 --> 00:01:40.430 Like that. 00:01:40.430 --> 00:01:41.070 Then I'm going to do it. 00:01:41.070 --> 00:01:43.170 And then I'm just going to try to continue the line from here. 00:01:43.170 --> 00:01:47.400 That might be the best technique. 00:01:47.400 --> 00:01:48.293 Something like that. 00:01:57.680 --> 00:01:58.570 So, let's look at that line. 00:01:58.570 --> 00:02:02.270 So what we want to do in this presentation is, figure out 00:02:02.270 --> 00:02:04.000 the slope of that line. 00:02:04.000 --> 00:02:06.225 So let's write out a couple of things that 00:02:06.225 --> 00:02:07.120 I think will help you. 00:02:07.120 --> 00:02:09.410 So, there's a couple ways to view slope. 00:02:09.410 --> 00:02:11.930 I think, intuitively, you know that the slope is the 00:02:11.930 --> 00:02:13.170 inclination of this line. 00:02:13.170 --> 00:02:14.470 And we can already see that this is a 00:02:14.470 --> 00:02:15.880 downward sloping line. 00:02:15.880 --> 00:02:18.820 Because it comes from the top left to the bottom right. 00:02:18.820 --> 00:02:20.590 So it's going to be a negative number, the slope. 00:02:20.590 --> 00:02:22.200 So you know that immediately. 00:02:22.200 --> 00:02:24.770 And we'll have -- what we're going to do is figure out how 00:02:24.770 --> 00:02:26.750 to figure out the slope. 00:02:26.750 --> 00:02:32.280 So the slope, let me write this down, slope and -- oftentimes 00:02:32.280 --> 00:02:35.700 they'll use the variable m, for slope, I have no idea why. 00:02:35.700 --> 00:02:39.060 Because m, clearly, does not stand for slope. 00:02:39.060 --> 00:02:41.370 That is equal to -- there's a couple of things 00:02:41.370 --> 00:02:42.300 you might hear. 00:02:42.300 --> 00:02:45.270 Change in y over change in x. 00:02:45.270 --> 00:02:48.920 That triangle, which is pronounced, delta just a Greek 00:02:48.920 --> 00:02:50.560 letter, that means change. 00:02:50.560 --> 00:02:52.560 The change in y over change in x. 00:02:52.560 --> 00:02:57.970 And that also is equal to rise over run. 00:02:57.970 --> 00:02:59.800 And I'm going to explain what all of this means in a second. 00:02:59.800 --> 00:03:01.650 So let's start at one of these points. 00:03:01.650 --> 00:03:05.130 Let's start at this green point, negative 1, 3. 00:03:05.130 --> 00:03:09.760 So how much do we have to rise and how much do we have to run 00:03:09.760 --> 00:03:12.710 to get to the second point, 2, 1? 00:03:12.710 --> 00:03:14.130 So let's do the rise first. 00:03:14.130 --> 00:03:21.640 Well, we have to go minus 2, so that's the rise. 00:03:21.640 --> 00:03:25.400 So the rise is equal to minus 2. 00:03:25.400 --> 00:03:28.120 Because we have to go down 2 to get to the same y 00:03:28.120 --> 00:03:29.290 as this yellow point. 00:03:29.290 --> 00:03:33.450 And then we have to run right there. 00:03:33.450 --> 00:03:36.830 We have to run plus 3. 00:03:36.830 --> 00:03:42.140 So rise divided by run is equal to minus 2 over 3. 00:03:42.140 --> 00:03:44.340 Well, how would we do that if we didn't have this nice graph 00:03:44.340 --> 00:03:46.790 here to actually draw on? 00:03:46.790 --> 00:03:51.400 Well, what we can do is, we can say let's take this 00:03:51.400 --> 00:03:53.690 as a starting point. 00:03:53.690 --> 00:04:00.360 Change in y, change in y, over change in x, is equal to 00:04:00.360 --> 00:04:04.110 we take the first y point, which is 3. 00:04:04.110 --> 00:04:06.010 And we subtract the second y point, which 00:04:06.010 --> 00:04:07.700 is 1, you see that? 00:04:07.700 --> 00:04:10.590 We just took 3 minus 1. 00:04:10.590 --> 00:04:16.620 So that's the change in y over, and we take the first x point. 00:04:16.620 --> 00:04:22.250 Negative 1, minus the second x point, minus 00:04:22.250 --> 00:04:25.345 2, so 3 minus 1 is 2. 00:04:25.345 --> 00:04:30.590 And negative 1 minus 2 is equal to minus 3. 00:04:30.590 --> 00:04:30.990 So, same thing. 00:04:30.990 --> 00:04:33.880 We got minus 2 over 3. 00:04:33.880 --> 00:04:35.070 Now we could have done it the other way. 00:04:35.070 --> 00:04:36.850 And I'm running out of space here. 00:04:36.850 --> 00:04:41.380 But we could've made this the first point. 00:04:41.380 --> 00:04:43.770 If we made that the first point, then the change in y 00:04:43.770 --> 00:04:47.100 would have been -- I want to make it really cluttered, 00:04:47.100 --> 00:04:48.460 so to confuse you. 00:04:48.460 --> 00:04:50.300 Change in y would be this y. 00:04:50.300 --> 00:04:57.380 1 minus 3 over change in x, would be 2, minus minus 1. 00:04:57.380 --> 00:05:00.960 Well, 1 minus 3 is minus 2. 00:05:00.960 --> 00:05:03.330 And 2 minus negative 1 is 3. 00:05:03.330 --> 00:05:06.620 So, once again, we got minus 2/3, So it doesn't matter which 00:05:06.620 --> 00:05:10.030 point we start with, as long as, if we use the y in this 00:05:10.030 --> 00:05:12.310 coordinate first, then we have to use the x in that 00:05:12.310 --> 00:05:13.480 coordinate first. 00:05:13.480 --> 00:05:14.900 Let's do some more problems. 00:05:14.900 --> 00:05:17.280 Actually, I'm going to do a couple just so you see the 00:05:17.280 --> 00:05:19.565 algebra without even graphing it first. 00:05:22.450 --> 00:05:24.560 So, let's say I wanted to figure out the slope between 00:05:24.560 --> 00:05:33.100 the points 5, 2, and 3, 5. 00:05:33.100 --> 00:05:35.760 Well, let's take this as our starting point. 00:05:35.760 --> 00:05:40.930 So, change in y over change in x, or rise over run, well, 00:05:40.930 --> 00:05:43.430 change in y would be this 5. 00:05:43.430 --> 00:05:46.950 5 minus this 2. 00:05:46.950 --> 00:05:52.471 Over this 3 minus this 5. 00:05:52.471 --> 00:05:59.110 And that gets us 3, this is a 5, over minus 2. 00:05:59.110 --> 00:06:01.890 Equals minus 3/2. 00:06:01.890 --> 00:06:04.370 Let's do another one. 00:06:04.370 --> 00:06:06.070 This time I'm going to try to make it color-coded so it'll 00:06:06.070 --> 00:06:08.070 more self-explanatory. 00:06:08.070 --> 00:06:09.410 Say, it's 1, 2. 00:06:09.410 --> 00:06:10.990 That's the first point. 00:06:10.990 --> 00:06:17.390 And then the second point is 4, 3. 00:06:17.390 --> 00:06:25.120 So, once again, we say slope is equal to change in 00:06:25.120 --> 00:06:28.790 y over change in x. 00:06:28.790 --> 00:06:29.980 Well, in y. 00:06:29.980 --> 00:06:31.180 We take the first y. 00:06:31.180 --> 00:06:32.270 Let's start here. 00:06:32.270 --> 00:06:33.850 And we'll call that y1. 00:06:33.850 --> 00:06:42.250 So that's 3 minus the second y, which is that 2. 00:06:42.250 --> 00:06:47.240 And then all of that over, once again, the first x. 00:06:47.240 --> 00:06:54.045 Which is 4, minus the second x, which is that 1. 00:06:54.045 --> 00:07:00.230 And this equals 3 minus 2, is 1. 00:07:00.230 --> 00:07:02.620 And 4 minus 1 is 3. 00:07:02.620 --> 00:07:05.540 So the slope in this example is 1/3. 00:07:05.540 --> 00:07:06.860 And we could have actually switched it around. 00:07:06.860 --> 00:07:08.440 We could have also done it other way. 00:07:08.440 --> 00:07:22.380 We could have said, 2 minus 3 over 1 minus 4. 00:07:22.380 --> 00:07:24.820 In which case we would have gotten negative 00:07:24.820 --> 00:07:26.800 1 over negative 3. 00:07:26.800 --> 00:07:28.170 Well, that just equals 1/3 again. 00:07:28.170 --> 00:07:29.750 Because the negatives cancel out. 00:07:29.750 --> 00:07:32.660 So I'll let you think about why this and this come 00:07:32.660 --> 00:07:34.110 out to the same thing. 00:07:34.110 --> 00:07:36.810 But the important thing to realize is, if we use the 3 00:07:36.810 --> 00:07:40.290 first, if we use the 3 first for the y, we also have to 00:07:40.290 --> 00:07:42.320 use the 4 first for the x. 00:07:42.320 --> 00:07:43.550 That's a common mistake. 00:07:43.550 --> 00:07:45.810 And also, you always have to be very careful with the negative 00:07:45.810 --> 00:07:48.240 signs when you do these type of problems. 00:07:48.240 --> 00:07:51.240 But I think that will give you at least enough of a sense that 00:07:51.240 --> 00:07:53.740 you could start the slope problems. 00:07:53.740 --> 00:07:55.450 The next module, I'll actually show you how to figure 00:07:55.450 --> 00:07:56.380 out the y intercept. 00:07:56.380 --> 00:07:59.250 Because, as we said, before the equation of any line is, 00:07:59.250 --> 00:08:02.880 y is equal to m x plus b. 00:08:02.880 --> 00:08:04.680 And I'm going to go into some more detail. 00:08:04.680 --> 00:08:06.280 Where m is the slope. 00:08:06.280 --> 00:08:08.310 So if you know the slope of a line. 00:08:08.310 --> 00:08:11.403 And you know the y intercept of a line, you know everything you 00:08:11.403 --> 00:08:13.350 need to know about the line, and you can actually write down 00:08:13.350 --> 00:08:15.180 the equation of a line, and figure out other points 00:08:15.180 --> 00:08:15.990 that are on it. 00:08:15.990 --> 00:08:18.400 So I'm going to do that in future modules. 00:08:18.400 --> 00:08:20.660 I hope I haven't confused you too much. 00:08:20.660 --> 00:08:22.620 And try some of those the slope modules. 00:08:22.620 --> 00:08:23.410 You should be able to do them. 00:08:23.410 --> 00:08:25.710 And I hope you have fun.

Algebra: Slope and Y-intercept intuition

https://www.youtube.com/watch?v=Nhn-anmubYU

vtt

https://www.youtube.com/api/timedtext?v=Nhn-anmubYU&ei=g2eUZcuoCZ_QxN8Prsua8A8&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=A7DB14747F9F53C29CDD0811CF78821300E1D56A.A761CCEDF35671B04A30A389D5CB5ED763C47D70&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:02.120 --> 00:00:02.770 Good morning. 00:00:02.770 --> 00:00:04.660 Actually I don't know what time it is for you, 00:00:04.660 --> 00:00:05.680 it's morning for me. 00:00:05.680 --> 00:00:09.490 Welcome to the presentation on slope and y-intercept. 00:00:09.490 --> 00:00:12.740 This presentation isn't going to teach you how to solve for 00:00:12.740 --> 00:00:15.760 slope and y-intercept, but hopefully it will give you 00:00:15.760 --> 00:00:18.660 a good intuition of what slope and y-intercept is. 00:00:18.660 --> 00:00:21.380 And we're going to do something a little bit different 00:00:21.380 --> 00:00:22.800 this time as opposed to what we normally do. 00:00:22.800 --> 00:00:23.880 We're not going to use the chalkboard, we're actually 00:00:23.880 --> 00:00:27.490 going to go on to the Khan academy website and use the 00:00:27.490 --> 00:00:30.210 graph of a line exercise to get a little bit of an intuition 00:00:30.210 --> 00:00:32.810 for what slope and y-intercept is. 00:00:32.810 --> 00:00:35.420 So when the application starts off, it starts with the 00:00:35.420 --> 00:00:38.360 equation y equals 1x plus 1. 00:00:38.360 --> 00:00:40.820 So that's the same thing as y equals x plus 1. 00:00:40.820 --> 00:00:42.790 But we see that the slope here is 1. 00:00:42.790 --> 00:00:44.580 If you looked at the introduction to graphing that 00:00:44.580 --> 00:00:46.430 I talked about, the slope is the same thing is the 00:00:46.430 --> 00:00:49.310 coefficient on the x term. 00:00:49.310 --> 00:00:52.910 And if you see here, whenever we move over 00:00:52.910 --> 00:00:55.600 by 1, we move up by 1. 00:00:55.600 --> 00:00:57.710 And I'm going to do another module on that slope is 00:00:57.710 --> 00:01:00.070 actually rise over run. 00:01:00.070 --> 00:01:03.510 So it's for every amount you rise, how much do you 00:01:03.510 --> 00:01:04.990 have to run to get that? 00:01:04.990 --> 00:01:07.940 And rise just means how much do you change in y, run means 00:01:07.940 --> 00:01:09.310 how much do you change in x. 00:01:09.310 --> 00:01:14.070 So here rise over run is just 1, and y-intercept is where 00:01:14.070 --> 00:01:16.550 you intercept the y-axis. 00:01:16.550 --> 00:01:20.160 Now, as I change the slope and the y-intercept for this graph, 00:01:20.160 --> 00:01:21.770 I think it's going to make a little bit more sense to you. 00:01:21.770 --> 00:01:25.440 Watch what happens when the slope goes from 1 to 3/2. 00:01:25.440 --> 00:01:27.970 So 3/2 is the same thing is 1 and 1/2. 00:01:27.970 --> 00:01:29.830 So notice it got steeper. 00:01:29.830 --> 00:01:32.460 And if I increase the slope more it gets steeper even 00:01:32.460 --> 00:01:35.530 more, and y equals 2x. 00:01:35.530 --> 00:01:37.960 If I increase any more 5/2 is 2 and 1/2. 00:01:37.960 --> 00:01:40.200 So the more I increase the slope, I think you 00:01:40.200 --> 00:01:40.940 see what's happening. 00:01:40.940 --> 00:01:42.380 This thing jumps around. 00:01:42.380 --> 00:01:44.030 I should fix that. 00:01:44.030 --> 00:01:45.100 Let me move it back. 00:01:45.100 --> 00:01:50.400 And actually, the goal is to make the line go through 00:01:50.400 --> 00:01:51.490 those two blue points. 00:01:51.490 --> 00:01:54.900 That's the goal of I guess you'd call it the game. 00:01:54.900 --> 00:01:57.070 I don't like how this thing jumps around though. 00:01:57.070 --> 00:01:58.835 That was interesting, let me go back there. 00:02:01.420 --> 00:02:04.460 y equals 0x plus 1. 00:02:04.460 --> 00:02:07.800 We could have rewritten this as just y equals 1, because 0x 00:02:07.800 --> 00:02:08.700 is the same things as 0. 00:02:08.700 --> 00:02:10.730 And notice it's a completely flat line. 00:02:10.730 --> 00:02:13.190 No matter what x is y is 1. 00:02:13.190 --> 00:02:14.580 And that makes sense because this equation would 00:02:14.580 --> 00:02:17.750 just be y equals 1. 00:02:17.750 --> 00:02:20.190 Now I've been showing you what happens to the slope. 00:02:20.190 --> 00:02:22.395 Now notice we have a negative slope. 00:02:22.395 --> 00:02:24.560 The slope is now downwards sloping. 00:02:24.560 --> 00:02:27.610 It's downward sloping at a slope of 1/2. 00:02:27.610 --> 00:02:31.020 Because let's say the rise in this situation is negative 00:02:31.020 --> 00:02:32.810 1, and the run is 2. 00:02:32.810 --> 00:02:36.500 So that's why we get negative 1 over 2. 00:02:36.500 --> 00:02:38.880 And we had just been doing slope so far and I think you 00:02:38.880 --> 00:02:42.170 get the idea that as we decrease slope, it's going to 00:02:42.170 --> 00:02:45.640 push the line further and further-- it's going to 00:02:45.640 --> 00:02:46.720 slope downward even more. 00:02:46.720 --> 00:02:50.490 I hate to use a word in its own definition, but I think you 00:02:50.490 --> 00:02:52.030 see that now in the picture. 00:02:52.030 --> 00:02:53.520 Now let's put up the y-intercept a little bit. 00:02:53.520 --> 00:02:54.460 And this is even more interesting. 00:02:54.460 --> 00:02:59.980 So y-intercept-- oh boy, how did that happen, that was 00:02:59.980 --> 00:03:03.760 strange --y-intercept-- 00:03:03.760 --> 00:03:08.820 Notice, negative 1x plus 2, so the slope is negative 1 but it 00:03:08.820 --> 00:03:10.420 intersects the y-axis at 2. 00:03:10.420 --> 00:03:13.615 Now if we increase y-intercept by 1 more it's just going 00:03:13.615 --> 00:03:15.570 to push this line up 1. 00:03:15.570 --> 00:03:17.670 Let's do that. 00:03:17.670 --> 00:03:17.940 See. 00:03:17.940 --> 00:03:21.110 Oh, this is actually increasing it by increments of 1/2. 00:03:21.110 --> 00:03:23.690 Let's do another one, I just want to see what happens 00:03:23.690 --> 00:03:24.530 on another graph. 00:03:24.530 --> 00:03:27.670 It actually depends on the actual problem. 00:03:27.670 --> 00:03:29.350 OK, this is interesting. 00:03:29.350 --> 00:03:30.140 OK, this is the same thing. 00:03:30.140 --> 00:03:31.600 We start at the same point. 00:03:31.600 --> 00:03:37.030 Let's actually try to figure out the equation of a line that 00:03:37.030 --> 00:03:39.120 goes through these two points. 00:03:39.120 --> 00:03:40.150 Well, let's see. 00:03:40.150 --> 00:03:41.890 It looks like the y-intercept if is going to have to 00:03:41.890 --> 00:03:43.970 be a little bit lower. 00:03:43.970 --> 00:03:45.980 I do not get why it would do that. 00:03:49.770 --> 00:03:54.240 It just brings the line down as we lower the y-intercept. 00:03:54.240 --> 00:03:56.520 And let's see I think the slope needs to be higher, because 00:03:56.520 --> 00:03:58.100 those two points, the line that goes through them is 00:03:58.100 --> 00:04:00.270 definitely steeper. 00:04:00.270 --> 00:04:01.980 I apologize for this thing acting up like. 00:04:01.980 --> 00:04:04.530 That looks like about the right slope. 00:04:04.530 --> 00:04:08.320 The slope is like that, and these two points are connected. 00:04:08.320 --> 00:04:10.240 Yeah, I think that looks like the right slope, but the 00:04:10.240 --> 00:04:11.470 y-intercept has to be lower. 00:04:16.130 --> 00:04:17.875 Almost there, I think. 00:04:20.570 --> 00:04:21.690 There you go! 00:04:21.690 --> 00:04:24.970 So the equation of this line is 7/4x. 00:04:24.970 --> 00:04:28.410 So 7/4, that's the same thing as like 1.75. 00:04:28.410 --> 00:04:33.360 So the slope of this line slopes faster than 1/1 and 00:04:33.360 --> 00:04:34.540 you can kind of see that. 00:04:34.540 --> 00:04:36.300 I'll show you how to figure out all this, I just want to give 00:04:36.300 --> 00:04:38.980 you an intuitive sense of what sloping and y-intercept is. 00:04:38.980 --> 00:04:42.520 And it intersects the y-axis at negative 13/4. 00:04:42.520 --> 00:04:45.210 That's a little more than 3, which you can-- negative 00:04:45.210 --> 00:04:47.860 3 --which you can see right there. 00:04:47.860 --> 00:04:50.890 Let's see if we can do another one. 00:04:50.890 --> 00:04:53.120 And if you want, we can assign this module to you and you can 00:04:53.120 --> 00:04:55.380 play with it just like I'm doing right here. 00:04:55.380 --> 00:04:58.290 So let's see, the line that we want to get will go 00:04:58.290 --> 00:04:59.320 something like that. 00:04:59.320 --> 00:05:02.540 Looks like the current line's slope is a little too high. 00:05:02.540 --> 00:05:04.130 Let me lower the slope a little bit. 00:05:04.130 --> 00:05:05.420 That looks about right. 00:05:05.420 --> 00:05:09.140 7/8, so that means for every 8 you move to the right 00:05:09.140 --> 00:05:10.580 you're going to move 7 up. 00:05:10.580 --> 00:05:12.790 And I'm going to draw that better in another module. 00:05:12.790 --> 00:05:16.470 This module I'm kind of doing on the fly, so I apologize. 00:05:16.470 --> 00:05:18.920 I do every model on the fly so I guess I really 00:05:18.920 --> 00:05:20.390 should apologize. 00:05:20.390 --> 00:05:24.350 But you're not paying for this, so I shouldn't apologize. 00:05:24.350 --> 00:05:27.240 Oh, I distracted very easily. 00:05:27.240 --> 00:05:29.180 Let's see, let's move this line up. 00:05:29.180 --> 00:05:30.660 And you do that just by the y-intercept. 00:05:30.660 --> 00:05:32.860 You can see shifting the y-intercept up just shifts 00:05:32.860 --> 00:05:33.890 the line straight up. 00:05:33.890 --> 00:05:36.200 It doesn't change the inclination of the line. 00:05:36.200 --> 00:05:38.100 The slope changes the inclination of the line. 00:05:41.140 --> 00:05:42.190 There we go. 00:05:42.190 --> 00:05:46.390 The equation of this line is 7/8x plus 13/4. 00:05:46.390 --> 00:05:49.060 Let's see if what I said about slope is right if we move. 00:05:49.060 --> 00:05:51.640 If we run 8, we should rise 7. 00:05:51.640 --> 00:05:52.040 So let's see. 00:05:52.040 --> 00:05:53.080 Run 8. 00:05:53.080 --> 00:06:02.090 1, 2, 3, 4, 5, 6, 7, 8. 00:06:02.090 --> 00:06:03.790 So that gets us right there. 00:06:03.790 --> 00:06:06.670 And then we should rise 7. 00:06:06.670 --> 00:06:14.930 1, 2, 3, 4, 5, 6, 7. 00:06:14.930 --> 00:06:16.740 Well that actually gets us those exact points. 00:06:16.740 --> 00:06:18.250 And we're back on the line again. 00:06:18.250 --> 00:06:21.170 I'm going to draw another thing like that for you so if you get 00:06:21.170 --> 00:06:23.240 confused don't lose heart. 00:06:23.240 --> 00:06:26.370 Let's do one more. 00:06:26.370 --> 00:06:27.800 OK. 00:06:27.800 --> 00:06:29.090 Where's the other dot? 00:06:29.090 --> 00:06:31.560 I don't know. 00:06:31.560 --> 00:06:33.610 Let me see. 00:06:33.610 --> 00:06:34.870 The other dot doesn't exist. 00:06:34.870 --> 00:06:37.170 I gotta fix all these bugs in this thing. 00:06:37.170 --> 00:06:37.540 Oh there. 00:06:37.540 --> 00:06:37.830 Good. 00:06:37.830 --> 00:06:38.440 It showed up. 00:06:38.440 --> 00:06:39.020 It showed up. 00:06:39.020 --> 00:06:39.850 Excellent. 00:06:39.850 --> 00:06:41.090 OK, so look. 00:06:41.090 --> 00:06:43.530 We have to make the line go through these two points. 00:06:43.530 --> 00:06:45.910 It looks like the slope is negative, definitely. 00:06:45.910 --> 00:06:49.120 Not that negative, it's like a fractional negative slope. 00:06:49.120 --> 00:06:51.720 And it'll intercept the y-axis somewhere around here. 00:06:51.720 --> 00:06:54.400 The y-intercept is going to be like 7 and something. 00:06:54.400 --> 00:06:55.360 7 and change. 00:06:55.360 --> 00:06:56.910 So first of all let's get this slope down. 00:06:56.910 --> 00:06:57.110 Oh boy. 00:06:57.110 --> 00:07:00.410 This thing is going to jump around again. 00:07:00.410 --> 00:07:02.410 Notice y equals 0, x plus 1. 00:07:02.410 --> 00:07:05.020 If we increase the slope. 00:07:05.020 --> 00:07:07.300 This thing is doing all sorts-- I haven't seen this application 00:07:07.300 --> 00:07:11.130 in a while, so I must've written it when I had inferior 00:07:11.130 --> 00:07:15.180 programming skills, let me keep --OK, that slope 00:07:15.180 --> 00:07:16.160 might be right. 00:07:16.160 --> 00:07:17.960 Let's bring the line up higher. 00:07:21.550 --> 00:07:26.200 No, it still seems like my slope-- see the y-intercept, 00:07:26.200 --> 00:07:27.215 I'm raising the line. 00:07:30.160 --> 00:07:31.830 Oh good, I got it exactly right. 00:07:31.830 --> 00:07:32.620 And I was right. 00:07:32.620 --> 00:07:36.310 The slope is negative, because you can see it slopes downward. 00:07:36.310 --> 00:07:38.710 But it's not sloping downward that fast. 00:07:38.710 --> 00:07:41.130 And that make sense, that the slope is negative 1/3. 00:07:41.130 --> 00:07:49.850 And that makes sense because if we run 3, 1, 2, 3, we rise 00:07:49.850 --> 00:07:53.060 negative 1, we rise negative 1. 00:07:53.060 --> 00:07:53.540 Right there. 00:07:53.540 --> 00:07:55.940 So that's why the slope is negative 1/3. 00:07:55.940 --> 00:07:58.530 And then the y-intercept is 22/3. 00:07:58.530 --> 00:08:00.170 Well that's 7 and 1/3. 00:08:00.170 --> 00:08:03.650 And right there, we intercept the y-axis 1/3 of the 00:08:03.650 --> 00:08:06.850 way between 7 and 8. 00:08:06.850 --> 00:08:10.020 Well I think that should at least give you a little bit of 00:08:10.020 --> 00:08:14.600 an intuition on what slope and y-intercept are and you can 00:08:14.600 --> 00:08:16.840 have this module assigned for you, so you could play 00:08:16.840 --> 00:08:17.540 with it yourself. 00:08:17.540 --> 00:08:19.940 And I'm going to do some more models where you actually 00:08:19.940 --> 00:08:22.650 calculate slope and y-intercept and hopefully give you even 00:08:22.650 --> 00:08:24.820 though further intuition on what they are. 00:08:24.820 --> 00:08:28.850 So I hope you have fun playing around with this stuff. 00:08:28.850 --> 00:08:31.000 I remember I was very excited when I first learned 00:08:31.000 --> 00:08:32.890 this stuff, because it's very visual. 00:08:32.890 --> 00:08:34.870 So, have fun.

Algebra: graphing lines 1

https://www.youtube.com/watch?v=2UrcUfBizyw

vtt

https://www.youtube.com/api/timedtext?v=2UrcUfBizyw&ei=g2eUZbH_B__UxN8P_tWNoAg&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=795379D59D36D5359A3F54F5B1DB34CCB2886321.0A73AD0B0245FEE60FBB07F31E69A5AA36D0EA93&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.220 --> 00:00:04.090 Welcome to the presentation on graphing lines. 00:00:04.090 --> 00:00:06.200 Let's get started. 00:00:06.200 --> 00:00:10.530 So let's say I had the equation-- let me make sure 00:00:10.530 --> 00:00:14.170 that this line doesn't show up too thick. 00:00:14.170 --> 00:00:18.370 Let's say I had the equation-- why isn't that showing up? 00:00:18.370 --> 00:00:18.760 Let's see. 00:00:18.760 --> 00:00:19.990 Oh, there you go. 00:00:19.990 --> 00:00:26.692 y is equal to 2x plus 1. 00:00:26.692 --> 00:00:29.670 So this is giving a relationship between x and y. 00:00:29.670 --> 00:00:35.930 So say x equals 1, then y would be 2 times 1 plus 1 or 3. 00:00:35.930 --> 00:00:39.140 So for every x that we can think of we can think 00:00:39.140 --> 00:00:41.350 of a corresponding y. 00:00:41.350 --> 00:00:42.710 So let's do that. 00:00:42.710 --> 00:00:46.520 If we said that-- put a little table here. 00:00:49.850 --> 00:00:50.400 x and y. 00:00:50.400 --> 00:00:53.630 And let's just throw out some random numbers for x. 00:00:53.630 --> 00:00:58.940 If x was let's say, negative 1, then y would be 2 times 00:00:58.940 --> 00:01:00.940 negative 1, which is negative 2. 00:01:00.940 --> 00:01:04.810 Plus 1, which would be negative 1. 00:01:04.810 --> 00:01:07.480 If x was 0 that's easy. 00:01:07.480 --> 00:01:09.310 It'd be 2 times 0, which is 0. 00:01:09.310 --> 00:01:11.672 Plus 1, which is 1. 00:01:11.672 --> 00:01:17.830 If x was 1, y would be 2 times 1, which is 2. 00:01:17.830 --> 00:01:21.700 Plus 1, which is 3. 00:01:21.700 --> 00:01:26.520 If x was 2, then I think you get the idea here. 00:01:26.520 --> 00:01:27.730 y would be 5. 00:01:27.730 --> 00:01:29.530 And we could keep on going. 00:01:29.530 --> 00:01:31.540 Obviously, there are an infinite number of x's we 00:01:31.540 --> 00:01:33.820 could choose and we could pick a corresponding y. 00:01:33.820 --> 00:01:35.410 So now you see we have a little table that gives the 00:01:35.410 --> 00:01:37.470 relationships between x and y. 00:01:37.470 --> 00:01:40.540 What we can do now is actually graph those points on 00:01:40.540 --> 00:01:42.500 a coordinate axis. 00:01:42.500 --> 00:01:46.870 So let me see if I can draw this somewhat neatly. 00:01:46.870 --> 00:01:50.425 I'll use this line so I get straight lines. 00:01:55.670 --> 00:01:56.970 That's pretty good. 00:01:56.970 --> 00:02:00.905 Again, let me draw some coordinate points. 00:02:00.905 --> 00:02:08.210 So let's say that's 1, that's 2, that's 3. 00:02:08.210 --> 00:02:13.170 This is negative 1, negative 2, negative 3. 00:02:13.170 --> 00:02:14.180 So this is the x-axis. 00:02:16.950 --> 00:02:22.340 We have 1, 2, 3. 00:02:22.340 --> 00:02:24.020 Notice we could keep going. 00:02:24.020 --> 00:02:28.730 1, 2, 3, and this is the y-axis. 00:02:32.255 --> 00:02:35.990 And this would be 1, 2, 3, and so on. 00:02:35.990 --> 00:02:37.220 This would be negative 1. 00:02:37.220 --> 00:02:38.360 I think you get the idea. 00:02:38.360 --> 00:02:40.550 So we can graph each of these points. 00:02:40.550 --> 00:02:45.440 So if we have the point x is negative 1, y is negative 1. 00:02:45.440 --> 00:02:48.710 So x, we go along the x-axis here, and we go to x is 00:02:48.710 --> 00:02:49.745 equal to negative 1. 00:02:49.745 --> 00:02:52.990 Then we go to y is equal to negative 1, so the point 00:02:52.990 --> 00:02:53.775 would be right here. 00:02:53.775 --> 00:02:56.520 Hope that makes sense to you. 00:02:56.520 --> 00:02:57.670 That's the point. 00:02:57.670 --> 00:03:00.650 I'll label it: negative 1 comma negative 1. 00:03:00.650 --> 00:03:01.140 It's a little messy. 00:03:01.140 --> 00:03:02.920 That says negative 1 comma negative 1. 00:03:02.920 --> 00:03:05.330 That point I just x'ed right there. 00:03:05.330 --> 00:03:06.820 Let's do another one. 00:03:06.820 --> 00:03:07.880 That's this point. 00:03:07.880 --> 00:03:10.280 I'll do it in a different color this time. 00:03:10.280 --> 00:03:14.960 Let's say we had the point 0 comma 1. 00:03:14.960 --> 00:03:17.410 Well, x is 0, which is here. 00:03:17.410 --> 00:03:21.110 And y is 1, so that point is right there. 00:03:21.110 --> 00:03:21.885 Let's do one more. 00:03:21.885 --> 00:03:25.350 If we have the point 1 comma 3. 00:03:25.350 --> 00:03:33.140 Well, 1 comma 3, x is 1 and we have y is 3. 00:03:33.140 --> 00:03:34.730 So we have the point right there. 00:03:34.730 --> 00:03:36.750 Hope that's making sense for you. 00:03:36.750 --> 00:03:39.180 And we could keep graphing them, but I think you see here, 00:03:39.180 --> 00:03:41.550 and especially if I had drawn this a little bit neater, that 00:03:41.550 --> 00:03:43.090 these points are forming a line. 00:03:43.090 --> 00:03:46.980 Let me draw that line in. 00:03:46.980 --> 00:03:49.370 The line looks something like this. 00:03:53.810 --> 00:03:54.860 That's not a good line. 00:03:54.860 --> 00:03:56.280 Let me do it better than that. 00:03:56.280 --> 00:03:58.350 The line looks something like this. 00:04:03.340 --> 00:04:03.860 You see that? 00:04:03.860 --> 00:04:06.720 Well, that's actually a pretty bad line that I just drew. 00:04:06.720 --> 00:04:11.670 So it would be a line that goes through-- let me change tools. 00:04:11.670 --> 00:04:13.960 It'd be a line that goes through here, through 00:04:13.960 --> 00:04:16.960 here, and through here. 00:04:16.960 --> 00:04:19.010 I don't know if I'm making this clear at all. 00:04:19.010 --> 00:04:22.630 Let me make these points a little bit. 00:04:22.630 --> 00:04:24.420 You see the line will go through all of these points, 00:04:24.420 --> 00:04:27.200 but it will also go through the point 2 comma 5, which will 00:04:27.200 --> 00:04:30.800 be up here some place. 00:04:30.800 --> 00:04:34.600 For any x that you can think of, if you had x is equal to 00:04:34.600 --> 00:04:38.510 10,380,000,000 the corresponding y will 00:04:38.510 --> 00:04:39.770 also be on this line. 00:04:39.770 --> 00:04:44.180 So this pink line, and it keeps going on forever, that 00:04:44.180 --> 00:04:49.850 represents every possible combination of x's and y's that 00:04:49.850 --> 00:04:51.580 will satisfy this equation. 00:04:51.580 --> 00:04:53.770 And of course, x doesn't have to just be whole 00:04:53.770 --> 00:04:54.880 numbers or integers. 00:04:54.880 --> 00:04:59.670 x could be pi-- 3.14159. 00:04:59.670 --> 00:05:02.090 In which case it would be someplace here and in which 00:05:02.090 --> 00:05:05.240 case y would be 2 pi plus 1. 00:05:05.240 --> 00:05:09.130 So every number that x could be there's a corresponding y. 00:05:09.130 --> 00:05:09.990 Let's do another 1. 00:05:14.060 --> 00:05:21.535 So if I had the equation y is equal to-- that's an ugly y. y 00:05:21.535 --> 00:05:29.020 is equal to negative 3x plus 5. 00:05:29.020 --> 00:05:32.180 Well, I'm going to draw it quick and dirty this time. 00:05:32.180 --> 00:05:34.180 So that's the x-axis. 00:05:34.180 --> 00:05:36.020 That's the y-axis. 00:05:36.020 --> 00:05:39.334 Let's put some values here. 00:05:39.334 --> 00:05:41.630 x and y. 00:05:41.630 --> 00:05:44.860 Let's say if x is negative 1, then negative 1 times 00:05:44.860 --> 00:05:49.245 negative 3 is 3 plus y is 8. 00:05:49.245 --> 00:05:52.750 If x is 0, then y is 5. 00:05:52.750 --> 00:05:54.020 That's pretty easy. 00:05:54.020 --> 00:05:58.690 If x is 1, negative 3 times 1 is negative 3. 00:05:58.690 --> 00:06:00.620 Then y is 2. 00:06:00.620 --> 00:06:04.990 If x is 2, negative 3 times 2 is negative 6. 00:06:04.990 --> 00:06:06.730 Then y is 1. 00:06:06.730 --> 00:06:08.070 Is that right? 00:06:08.070 --> 00:06:09.610 Negative 6-- no, no. 00:06:09.610 --> 00:06:10.315 Negative 1. 00:06:10.315 --> 00:06:12.810 I knew something was wrong there. 00:06:12.810 --> 00:06:14.610 So let's graph some of these points. 00:06:14.610 --> 00:06:18.000 So when x is negative 1 and I'm just kind of approximating. 00:06:18.000 --> 00:06:21.710 When x is negative 1, y is negative 8. 00:06:21.710 --> 00:06:23.610 So that point would be someplace around here. 00:06:23.610 --> 00:06:25.680 And there's a whole module I'm graphing coordinates if you're 00:06:25.680 --> 00:06:28.540 finding the graphing a coordinate pair to be 00:06:28.540 --> 00:06:31.240 a little confusing. 00:06:31.240 --> 00:06:31.700 Oh, wait. 00:06:31.700 --> 00:06:32.790 I just made a mistake. 00:06:32.790 --> 00:06:34.830 When x is negative 1, y is 9. 00:06:34.830 --> 00:06:37.070 Not negative 8, so ignore this right here. 00:06:37.070 --> 00:06:42.270 When x is negative 1, y is positive 8. 00:06:42.270 --> 00:06:44.610 So y being up here someplace. 00:06:44.610 --> 00:06:47.540 When x is 0, y is 5. 00:06:47.540 --> 00:06:50.440 So it'd be here someplace. 00:06:50.440 --> 00:06:53.720 When x is 1, y is 2. 00:06:53.720 --> 00:06:54.650 So it's like here. 00:06:57.160 --> 00:07:02.310 When x is 2, y is negative 1. 00:07:02.310 --> 00:07:04.550 So as you can see-- and I've approximated it. 00:07:04.550 --> 00:07:08.740 If I had graphing paper or if I had a better drawn chart you 00:07:08.740 --> 00:07:11.210 could have seen it and it would have been exactly right. 00:07:11.210 --> 00:07:14.950 I think this line will do the job. 00:07:14.950 --> 00:07:19.430 That every point that satisfies this equation actually 00:07:19.430 --> 00:07:21.470 falls on this line. 00:07:21.470 --> 00:07:23.700 And something interesting here I'll point out. 00:07:23.700 --> 00:07:26.790 You notice that this line it slopes downwards. 00:07:26.790 --> 00:07:29.510 It goes from the top left to the bottom right. 00:07:29.510 --> 00:07:31.800 While the line we had drawn before had gone from the 00:07:31.800 --> 00:07:35.160 bottom left to the top right. 00:07:35.160 --> 00:07:37.740 Is there anything about this equation that seems a little 00:07:37.740 --> 00:07:40.320 bit different than the last? 00:07:40.320 --> 00:07:43.240 I'll give you a little bit of a hint. 00:07:43.240 --> 00:07:47.230 This number-- the negative 3, or you could say that the 00:07:47.230 --> 00:07:52.280 coefficient on x-- that determines whether the line 00:07:52.280 --> 00:07:55.040 slopes upward, or the line slows downward, and it tells 00:07:55.040 --> 00:07:56.725 you also how steep the line is. 00:07:56.725 --> 00:07:58.900 And that actually, negative 3 is the slope. 00:07:58.900 --> 00:08:02.350 And I'm going to do a whole nother module on slope. 00:08:02.350 --> 00:08:05.400 And this number here is called the y-intercept. 00:08:05.400 --> 00:08:06.960 And that actually tells you where you're going 00:08:06.960 --> 00:08:08.970 to intersect the y-axis. 00:08:08.970 --> 00:08:10.460 And it turns out here, that you intersect the 00:08:10.460 --> 00:08:13.250 axis at 0 comma 5. 00:08:15.920 --> 00:08:18.470 Let's do one more real fast. 00:08:21.770 --> 00:08:26.130 y is equal to 2-- we already did 2x. 00:08:26.130 --> 00:08:35.374 y is equal to 1/2 x plus 2 So real fast. 00:08:35.374 --> 00:08:37.180 x and y. 00:08:37.180 --> 00:08:39.270 And you only need two points for a line, really. 00:08:39.270 --> 00:08:41.120 So you could just say let's say, x equals 0. 00:08:41.120 --> 00:08:43.410 That's easy. y equals 2. 00:08:43.410 --> 00:08:46.580 And if x equals 2 then y equals 3. 00:08:46.580 --> 00:08:51.660 So before when we were doing 3 and 4 points that was just to 00:08:51.660 --> 00:08:53.650 kind of show you, but you really just need two 00:08:53.650 --> 00:08:54.230 points for a line. 00:08:54.230 --> 00:08:57.930 So 0 comma 1 2. 00:08:57.930 --> 00:08:58.730 So that's on there. 00:08:58.730 --> 00:09:03.320 And then 1, 2 comma 3. 00:09:03.320 --> 00:09:05.950 So it's there. 00:09:05.950 --> 00:09:08.110 So the line is going to look something like this. 00:09:12.190 --> 00:09:14.440 So notice here, once again, we're upward sloping and that's 00:09:14.440 --> 00:09:16.870 because this 1/2 is positive. 00:09:16.870 --> 00:09:20.120 But we're not sloping-- we're not moving up as quickly as 00:09:20.120 --> 00:09:22.870 when we had y equals 2x. y equals 2x looked 00:09:22.870 --> 00:09:24.452 something like this. 00:09:24.452 --> 00:09:26.390 It was sloping up much, much, much faster. 00:09:26.390 --> 00:09:27.620 I hope I'm not confusing you. 00:09:27.620 --> 00:09:30.710 And then the y intercept of course is at 0 comma 2, 00:09:30.710 --> 00:09:32.160 which is right here. 00:09:32.160 --> 00:09:35.300 So if you ever want to graph a line it's really easy. 00:09:35.300 --> 00:09:37.760 You have to just try out some points and you can graph it. 00:09:37.760 --> 00:09:39.440 And now in the next module I'm going to show you a little bit 00:09:39.440 --> 00:09:41.360 more about slope and y-intercept and you won't 00:09:41.360 --> 00:09:42.480 even have to do this. 00:09:42.480 --> 00:09:45.490 But this gives you good intuitive feel, I think, 00:09:45.490 --> 00:09:47.320 what a graph of a line is. 00:09:47.320 --> 00:09:49.250 I hope you have fun.

Age word problems 2

https://www.youtube.com/watch?v=pPqPj8CAPvI

vtt

https://www.youtube.com/api/timedtext?v=pPqPj8CAPvI&ei=g2eUZYn5AYOsvdIP3a6fsAc&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=51AA9617C090503D8A68156D2DC3CB7914866711.8C967E2E389EDB5AD9E2B5C52AA8E3AFD9307B53&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.430 --> 00:00:06.220 Welcome to the second set of presentations 00:00:06.220 --> 00:00:08.640 on age word problems. 00:00:08.640 --> 00:00:10.200 I have one typed out here. 00:00:10.200 --> 00:00:11.240 Let's see what it says. 00:00:11.240 --> 00:00:15.530 It says, Salman- that's me- is 108 years old. 00:00:15.530 --> 00:00:19.500 And so this is clearly at some vast time in the future. 00:00:19.500 --> 00:00:23.380 And this'll only happen if my caloric restriction works out. 00:00:23.380 --> 00:00:26.370 But Salman is 108 years old. 00:00:26.370 --> 00:00:29.220 Jonathan is 24 years old. 00:00:29.220 --> 00:00:33.210 How many years will it take for Salman to be exactly 4 00:00:33.210 --> 00:00:35.930 times as old as Jonathan? 00:00:35.930 --> 00:00:38.720 Well, let's figure this one out. 00:00:38.720 --> 00:00:41.240 What we're trying to solve for is how many years will it take. 00:00:41.240 --> 00:00:43.740 So let's use the variable y, for years. 00:00:43.740 --> 00:00:53.880 So y equals years until, let's just say Sal, for short. 00:00:53.880 --> 00:00:57.275 Sal is 4 times Jonathan's age. 00:01:02.910 --> 00:01:09.275 Well, if today, I, or Sal, if Salman, it's hard to speak 00:01:09.275 --> 00:01:10.430 in the third person. 00:01:10.430 --> 00:01:16.840 If Salman is 100 years old today, in y years, Salman 00:01:16.840 --> 00:01:25.950 will be 108 plus y is equal to Sal in y years. 00:01:25.950 --> 00:01:28.920 We'll say y years. 00:01:28.920 --> 00:01:32.670 Sal in y years is going to be 108 plus y. 00:01:32.670 --> 00:01:35.330 And then Jonathan in y years, that's pretty easy. 00:01:35.330 --> 00:01:39.720 He's going to be 24 plus y. 00:01:39.720 --> 00:01:42.690 So that's, I'll say Jon for short. 00:01:42.690 --> 00:01:45.440 In y years. 00:01:45.440 --> 00:01:47.870 Not years. 00:01:47.870 --> 00:01:49.180 y years. 00:01:49.180 --> 00:01:51.260 And what else does the problem say? 00:01:51.260 --> 00:01:54.760 It says, in how many years would take for Salman to be 00:01:54.760 --> 00:01:57.610 exactly-- I should put that in another color for emphasis-- 00:01:57.610 --> 00:02:02.900 Exactly 4 times as old as Jonathan? 00:02:02.900 --> 00:02:04.770 Well, and the exactly is important. 00:02:04.770 --> 00:02:08.220 Because Salman is already more than 4 times 00:02:08.220 --> 00:02:09.340 as old as Jonathan. 00:02:09.340 --> 00:02:12.590 But they want to figure out exactly when is Salman going to 00:02:12.590 --> 00:02:14.890 be 4 times as old as Jonathan. 00:02:14.890 --> 00:02:21.450 Well, in y years Salman is going to be 108 plus y. 00:02:21.450 --> 00:02:22.190 So we know that. 00:02:22.190 --> 00:02:25.810 108 plus y. 00:02:25.810 --> 00:02:28.490 And after y years, he's going to be 4 times 00:02:28.490 --> 00:02:29.680 as old as Jonathan. 00:02:29.680 --> 00:02:31.460 Exactly 4 times. 00:02:31.460 --> 00:02:35.740 And Jonathan's going to be 24 plus y years old. 00:02:35.740 --> 00:02:37.690 And now we just solve the equation. 00:02:37.690 --> 00:02:43.510 Get 108 plus y is equal to 4 times 24. 00:02:43.510 --> 00:02:45.300 Well, 4 times 25 is 100. 00:02:45.300 --> 00:02:46.670 And we can subtract 4 from there. 00:02:46.670 --> 00:02:49.120 So it's 96. 00:02:49.120 --> 00:02:51.390 Plus 4y. 00:02:51.390 --> 00:02:54.030 And now we just solve this equation. 00:02:54.030 --> 00:02:56.820 We get 3. 00:02:56.820 --> 00:02:58.990 3, I'm going to skip some steps, because I think this 00:02:58.990 --> 00:03:00.390 part is easy for you. 00:03:00.390 --> 00:03:05.910 3y will equal 108 minus 96. 00:03:05.910 --> 00:03:13.950 So we get 3y is equal to, what is that, 12, right? 00:03:13.950 --> 00:03:16.780 So y is equal to 4. 00:03:16.780 --> 00:03:21.635 So our algebra has told us that in 4 years Salman is going 00:03:21.635 --> 00:03:24.260 to be exactly 4 times as old as Jonathan. 00:03:24.260 --> 00:03:25.630 Let's see if that's true. 00:03:25.630 --> 00:03:28.480 Well, if Salman is 108 right now, in 4 years 00:03:28.480 --> 00:03:30.780 he's going to be 112. 00:03:30.780 --> 00:03:33.900 And if Jonathan is 24 right now, in 4 years he's 00:03:33.900 --> 00:03:36.610 going to be 28. 00:03:36.610 --> 00:03:37.540 And let's see. 00:03:37.540 --> 00:03:41.780 28 times 4 is 80 plus 32. 00:03:41.780 --> 00:03:42.730 Yep, exactly. 00:03:42.730 --> 00:03:43.990 He'll be 112. 00:03:43.990 --> 00:03:45.780 Looks like that problem worked. 00:03:45.780 --> 00:03:46.920 Excellent. 00:03:46.920 --> 00:03:50.035 Let's do another one. 00:03:50.035 --> 00:03:53.520 Hope you didn't hear that, it was my stomach growling. 00:03:53.520 --> 00:03:55.240 See how hard I work on this site? 00:03:55.240 --> 00:03:56.220 Don't even eat properly. 00:03:59.400 --> 00:04:02.800 Let's do another problem. 00:04:02.800 --> 00:04:04.540 I'll type it in green this time. 00:04:07.040 --> 00:04:23.670 Tarush is 5 times as old as Arman is today, 85 years ago. 00:04:23.670 --> 00:04:26.620 We're dealing with huge swathes of time. 00:04:26.620 --> 00:04:29.920 85 years ago. 00:04:29.920 --> 00:04:39.190 Tarush was 10 times as old as Arman. 00:04:39.190 --> 00:04:45.730 How old is Arman today? 00:04:45.730 --> 00:04:47.970 Let's see if we can tackle this quite interesting 00:04:47.970 --> 00:04:50.090 problem, I think. 00:04:50.090 --> 00:04:51.640 Settings. 00:04:51.640 --> 00:04:53.970 Pen tool on. 00:04:53.970 --> 00:04:55.320 OK. 00:04:55.320 --> 00:04:57.380 Well, I think this might be useful to do one of those 00:04:57.380 --> 00:05:01.030 charts like we did in the first video presentation. 00:05:01.030 --> 00:05:03.980 Well, we're trying to solve for how old is Arman today's. 00:05:03.980 --> 00:05:05.750 So let's say Arman. 00:05:09.010 --> 00:05:09.700 And we're going to have today. 00:05:13.350 --> 00:05:14.600 And we're going to have, what's the other time 00:05:14.600 --> 00:05:15.470 period we're dealing with. 00:05:15.470 --> 00:05:17.020 We're dealing with 85 years ago. 00:05:17.020 --> 00:05:19.096 So, let's say. 00:05:19.096 --> 00:05:23.054 85 years ago. 00:05:23.054 --> 00:05:27.880 In a galaxy far, far away. 00:05:27.880 --> 00:05:34.890 OK, Arman and Tarush. 00:05:34.890 --> 00:05:36.970 Well, we're trying to solve for how old is Arman today's. 00:05:36.970 --> 00:05:39.280 So let's just make that x, for simplicity. 00:05:39.280 --> 00:05:43.150 It says Tarush is 5 times as old as Arman today. 00:05:43.150 --> 00:05:44.550 So let me underline that. 00:05:44.550 --> 00:05:48.310 Tarush is 5 times as old as Arman today. 00:05:48.310 --> 00:05:53.900 So if Arman is x, that tells us that Tarush is going to be 5x. 00:05:53.900 --> 00:05:56.330 85 years ago. 00:05:56.330 --> 00:06:03.210 So, 85 years ago, well, Arman is x today, then 85 years ago, 00:06:03.210 --> 00:06:05.980 Arman would have been x minus 85. 00:06:08.720 --> 00:06:16.230 And if Tarush is 5x today, then Tarush would be 5x minus 85. 00:06:16.230 --> 00:06:20.150 I just subtracted 85 from the current age. 00:06:20.150 --> 00:06:22.630 Because we're going 85 years in the past. 00:06:22.630 --> 00:06:25.480 And now we have this extra piece of information. 00:06:25.480 --> 00:06:30.500 Which tells us that 85 years ago, Tarush was 10 00:06:30.500 --> 00:06:32.420 times as old as Arman. 00:06:32.420 --> 00:06:37.940 So this number is going to be 10 times more than this number. 00:06:37.940 --> 00:06:39.340 That's what that sentence tells us. 00:06:39.340 --> 00:06:42.040 85 years ago, we're in this situation. 00:06:42.040 --> 00:06:46.330 Tarush, which is this, was 10 times older than Arman. 00:06:46.330 --> 00:06:48.580 We just write that out algebraically. 00:06:48.580 --> 00:06:54.230 85 years ago, Tarush is 5x minus 85. 00:06:54.230 --> 00:06:58.370 And the sentence tells us that he was 10 times older than 00:06:58.370 --> 00:07:02.620 Arman, who is x minus 85. 00:07:02.620 --> 00:07:04.390 Now we just solve the equation. 00:07:04.390 --> 00:07:14.900 We get 5x minus 85 is equal to 10x minus 850. 00:07:14.900 --> 00:07:17.900 And then you get-- and I'm going to do this, skip some 00:07:17.900 --> 00:07:24.830 steps, just to confuse you-- 5x is equal too-- well, I think 00:07:24.830 --> 00:07:26.370 I just confused myself. 00:07:26.370 --> 00:07:34.790 5x is equal to 850 minus 85. 00:07:34.790 --> 00:07:37.400 Let's see, what's 850 minus 85. 00:07:37.400 --> 00:07:41.230 It'll be 35 less than 800. 00:07:41.230 --> 00:07:46.500 So we could say that 5x-- 35 less than 800. 00:07:46.500 --> 00:07:48.250 30 less than 800 gets 770. 00:07:48.250 --> 00:07:52.070 So 765. 00:07:52.070 --> 00:07:54.590 And then x is equal to, let me see. 00:07:54.590 --> 00:07:58.520 5 goes into 70. 00:07:58.520 --> 00:08:01.140 1 times 5. 00:08:01.140 --> 00:08:03.260 26. 00:08:03.260 --> 00:08:08.230 5 times 5 is 25. 00:08:08.230 --> 00:08:09.586 15. 00:08:09.586 --> 00:08:13.810 So x is equal to 153. 00:08:13.810 --> 00:08:16.250 We have some long-lifespanned people. 00:08:16.250 --> 00:08:22.000 So we get the solution that Arman is 153 years old. 00:08:22.000 --> 00:08:23.190 Today. 00:08:23.190 --> 00:08:25.000 Let's see if that makes sense. 00:08:25.000 --> 00:08:31.300 Well, if he's 153, then, no, that can't be right. 00:08:31.300 --> 00:08:33.370 Because then he's 5. 00:08:33.370 --> 00:08:34.220 Huh. 00:08:34.220 --> 00:08:35.900 Well, actually I'm almost out of time. 00:08:35.900 --> 00:08:38.190 Let me see where I might have messed up on this. 00:08:38.190 --> 00:08:43.320 If Arman is x, Tarush is 5x. 00:08:43.320 --> 00:08:44.820 That makes sense. 00:08:44.820 --> 00:08:46.010 Oh, right, right, right. 00:08:46.010 --> 00:08:47.740 I think this could make sense. 00:08:47.740 --> 00:08:51.220 So, I thought that Tarush had to be younger than Arman. 00:08:51.220 --> 00:08:54.110 But no, Tarush is 5 times 153. 00:08:54.110 --> 00:08:57.630 So Tarush is actually 765 years old. 00:08:57.630 --> 00:08:59.400 We should call him Methuselah. 00:08:59.400 --> 00:09:04.190 So, Tarush is 765 years old. 00:09:04.190 --> 00:09:09.120 And then if we go 85 years into the past, Tarush would have 00:09:09.120 --> 00:09:11.040 been-- what did we say? 00:09:11.040 --> 00:09:15.760 If we go 85 years into the past, so let's write this down. 00:09:15.760 --> 00:09:18.820 This is equal to, I'm about to run out of time, so you might 00:09:18.820 --> 00:09:20.112 have to check this yourself. 00:09:20.112 --> 00:09:26.010 If that's equal to 153, then 5x is equal to 765. 00:09:26.010 --> 00:09:27.430 These numbers are big. 00:09:27.430 --> 00:09:30.700 And then 153 minus 85. 00:09:30.700 --> 00:09:35.710 Well, 155 minus 85 would be 73. 00:09:35.710 --> 00:09:39.840 So this would be 72. 00:09:39.840 --> 00:09:44.250 And then Tarush, 85 years ago. 00:09:44.250 --> 00:09:45.550 No, I think I messed up someplace. 00:09:48.670 --> 00:09:51.150 Well, actually, I only have ten seconds before YouTube 00:09:51.150 --> 00:09:52.130 won't let me upload it. 00:09:52.130 --> 00:09:54.450 So we'll have to stop there. 00:09:54.450 --> 00:09:55.950 Thank you.

Age word problems 3

https://www.youtube.com/watch?v=DplUpe3oyWo

vtt

https://www.youtube.com/api/timedtext?v=DplUpe3oyWo&ei=g2eUZe-jHrqfp-oP3e6tiAU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=92451278753E7C12A74C23373736B956F483BD96.38E3C7E5DB94B0BEE2523824E0858C157AA7BA99&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:02.170 --> 00:00:06.550 Welcome to the third in this series of age word problems. 00:00:06.550 --> 00:00:09.070 And I just wanted to start off where I left off 00:00:09.070 --> 00:00:10.210 in the second one. 00:00:10.210 --> 00:00:11.340 I got myself confused. 00:00:11.340 --> 00:00:13.370 You guys make me very nervous while you just sit there 00:00:13.370 --> 00:00:15.180 listening and I have to perform for you. 00:00:15.180 --> 00:00:18.180 But, it turns out we did get the right answer, if you say 00:00:18.180 --> 00:00:24.430 Arman is 153 and Tarush is 765, then 85 years ago Arman would 00:00:24.430 --> 00:00:28.990 have been 68 and Tarush would have been 680. 00:00:28.990 --> 00:00:31.720 And notice that's 10 times. 00:00:31.720 --> 00:00:33.040 So it worked out. 00:00:33.040 --> 00:00:35.380 So we're ready to tackle another problem. 00:00:35.380 --> 00:00:36.540 One more problem. 00:00:36.540 --> 00:00:37.210 Let's see. 00:00:49.140 --> 00:00:58.790 This problem says, Zack is 4 times as old as Salman. 00:00:58.790 --> 00:00:59.590 What a coincidence. 00:00:59.590 --> 00:01:02.560 Salman really shows up in a lot of problems. 00:01:02.560 --> 00:01:05.340 So Zack is 4 times as old as Salman. 00:01:05.340 --> 00:01:15.470 Zack is also 3 years older then Salman. 00:01:15.470 --> 00:01:19.490 How old is Zack? 00:01:19.490 --> 00:01:21.960 Well, actually this one actually seems fairly 00:01:21.960 --> 00:01:22.620 straightforward. 00:01:22.620 --> 00:01:24.830 We're not dealing with the present and the future. 00:01:24.830 --> 00:01:26.170 We're just dealing with the present. 00:01:26.170 --> 00:01:29.320 So let's see if we can figure this out. 00:01:29.320 --> 00:01:31.530 So we're trying to figure out how old is Zack. 00:01:31.530 --> 00:01:33.600 Let's say, z for Zack. 00:01:33.600 --> 00:01:35.125 z equals Zack. 00:01:40.460 --> 00:01:42.390 It says Zack is 4 times as old as Salman. 00:01:45.290 --> 00:01:49.990 So, Salman, if Zack is z, this first sentence, 00:01:49.990 --> 00:01:52.130 let me circle it. 00:01:52.130 --> 00:01:57.170 Zack is 4 times as old as Salman. 00:01:57.170 --> 00:02:03.740 So, if Zack is 4 times as old as Salman, that means that Sal 00:02:03.740 --> 00:02:19.020 is equal to Zack divided by 4, But then it also says here that 00:02:19.020 --> 00:02:23.940 Zack is also 3 years older than Salman. 00:02:23.940 --> 00:02:26.600 Well, if Zack is also 3 years older than Salman, so 00:02:26.600 --> 00:02:35.310 that means that also Sal would be Zack minus 3. 00:02:35.310 --> 00:02:37.510 The first sentence says Zack is 4 times as old as Salman. 00:02:37.510 --> 00:02:40.150 So that means if you take Zack's age and divide it 00:02:40.150 --> 00:02:42.110 by 4, you get Sal's age. 00:02:42.110 --> 00:02:45.290 The second sentence says Zack is also 3 years 00:02:45.290 --> 00:02:46.640 older than Salman. 00:02:46.640 --> 00:02:49.890 So that means if you take Zack's age and you subtract 00:02:49.890 --> 00:02:53.360 3 from it, you get Sal's age again. 00:02:53.360 --> 00:02:54.720 Well, we have our equation set up. 00:02:54.720 --> 00:02:56.190 And, actually, instead of writing Zack, I should 00:02:56.190 --> 00:02:57.440 have just written z. 00:02:57.440 --> 00:02:59.660 But the equation is all set up for us. 00:02:59.660 --> 00:03:05.500 We get z over 4 is equal to z minus 3. 00:03:05.500 --> 00:03:08.780 We can multiply both sides of this equation times 4 and we'll 00:03:08.780 --> 00:03:15.690 get z is equal to 4z, remember to distribute, minus 12. 00:03:15.690 --> 00:03:18.290 And then, we'll skip a couple of steps. 00:03:18.290 --> 00:03:22.030 And I get 3z is equal to 12. 00:03:22.030 --> 00:03:28.820 And I get Zack is equal to 4 years old. 00:03:28.820 --> 00:03:29.980 Let's see if that makes sense. 00:03:29.980 --> 00:03:33.460 If Zack is 4 years old, then this first sentence says Zack 00:03:33.460 --> 00:03:35.030 is 4 times as old as Salman. 00:03:35.030 --> 00:03:39.390 So that means Sal is 1 year old. 00:03:39.390 --> 00:03:41.700 It also says Zack is 3 years older than Salman. 00:03:41.700 --> 00:03:42.570 Well, that's consistent. 00:03:42.570 --> 00:03:44.580 4 is 3 more than 1. 00:03:44.580 --> 00:03:45.350 So we were right. 00:03:45.350 --> 00:03:48.400 Zack is exactly four years old. 00:03:48.400 --> 00:03:49.610 Hope that helps. 00:03:49.610 --> 00:03:50.090 Have fun. 00:03:50.090 --> 00:03:50.900 I think you're ready. 00:03:50.900 --> 00:03:53.820 You've now seen, at least, every type of 00:03:53.820 --> 00:03:55.000 the age word problems. 00:03:55.000 --> 00:03:56.485 If you're still a little confused, you might just want 00:03:56.485 --> 00:03:59.790 to re-watch the videos and maybe pause it right after I 00:03:59.790 --> 00:04:02.060 give you the problem, and see if you can solve it yourself. 00:04:02.060 --> 00:04:04.030 And remember, you had a lot of practice problems that you 00:04:04.030 --> 00:04:06.330 can do on the actual modules themselves. 00:04:06.330 --> 00:04:07.930 Have fun.

Age word problems 1

https://www.youtube.com/watch?v=bAUT_Pux73w

vtt

https://www.youtube.com/api/timedtext?v=bAUT_Pux73w&ei=g2eUZcGYFI6Dp-oP1LW4gAM&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=9117E37125D51F6595FF9D8BB005E98699E7A0A7.3F2FB9AAC8EAFB245B30029A93D62DAE27920F70&key=yt8&lang=en&name=English&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.190 --> 00:00:04.000 Welcome to the presentation on age word problems. 00:00:04.000 --> 00:00:04.860 Let's get started. 00:00:04.860 --> 00:00:06.800 Well, I already have one written down here. 00:00:06.800 --> 00:00:12.630 It says in 4 years, Ali will be 3 times as old as he is today. 00:00:12.630 --> 00:00:14.510 How old is Ali today? 00:00:14.510 --> 00:00:21.520 Well, let's just say that x -- that's a little too big for 00:00:21.520 --> 00:00:29.910 what I want to write -- x equals Ali's current age. 00:00:29.910 --> 00:00:32.170 Or his age today. 00:00:32.170 --> 00:00:33.170 Please forgive my handwriting. 00:00:35.740 --> 00:00:39.530 If x is Ali's current age -- and here we say, in 4 years. 00:00:39.530 --> 00:00:41.400 So how old is Ali going to be in 4 years? 00:00:41.400 --> 00:00:44.440 Well, if x is his current age, in 4 years, he's going to 00:00:44.440 --> 00:00:47.020 be x plus 4 years old. 00:00:47.020 --> 00:00:56.940 So that is equal to Ali's age in 4 years. 00:01:00.600 --> 00:01:02.155 Let's read what other information there 00:01:02.155 --> 00:01:02.600 is in the problem. 00:01:02.600 --> 00:01:04.560 It says, in 4 years. 00:01:04.560 --> 00:01:10.040 So in this situation, that's a 4, x plus 4, in 4 years 00:01:10.040 --> 00:01:13.150 Ali will be 3 times as old as he is today. 00:01:13.150 --> 00:01:14.750 Well, today he's x. 00:01:14.750 --> 00:01:17.116 So we say, in 4 years. 00:01:17.116 --> 00:01:19.140 x plus 4. 00:01:19.140 --> 00:01:22.520 And that's just -- so, in 4 years. 00:01:22.520 --> 00:01:26.360 So we could translate that into x plus 4. 00:01:26.360 --> 00:01:32.010 That's going to equal 3 times as old as he is today. 00:01:32.010 --> 00:01:33.210 Well, we know today he's x. 00:01:33.210 --> 00:01:36.820 So that's going to be equal to 3x. 00:01:36.820 --> 00:01:38.110 And now we just solve for x. 00:01:38.110 --> 00:01:39.350 Subtract x from both sides. 00:01:39.350 --> 00:01:41.530 You get 2x is equal to 4. 00:01:41.530 --> 00:01:43.280 x equals 2. 00:01:43.280 --> 00:01:47.100 So Ali is 2 years old today. 00:01:47.100 --> 00:01:49.990 Let's do another problem. 00:01:49.990 --> 00:01:53.340 That makes sense, because if he's 2 today, in 4 years 00:01:53.340 --> 00:01:54.500 he's going to be 6. 00:01:54.500 --> 00:01:56.890 Which is going to be 3 times as old as he is today. 00:01:56.890 --> 00:01:57.820 So it all works out. 00:01:57.820 --> 00:01:58.780 Let's do another problem. 00:02:01.400 --> 00:02:07.204 Let's clear this and now I'm going to attempt to type. 00:02:07.204 --> 00:02:08.660 Hope I don't mess up. 00:02:11.610 --> 00:02:12.110 OK. 00:02:12.110 --> 00:02:24.420 The problem, is Nazrith is 4 years older then Omama. 00:02:24.420 --> 00:02:36.470 2 years ago, Nazrith was 5 times as old as Omama. 00:02:39.230 --> 00:02:44.690 How old is Nazrith today? 00:02:44.690 --> 00:02:48.106 This one seems more difficult. 00:02:48.106 --> 00:02:51.820 But I think we'll be able to get through it. 00:02:51.820 --> 00:02:53.140 Let's figure it out. 00:02:53.140 --> 00:02:56.960 All right, so once again what are we trying to solve for. 00:02:56.960 --> 00:02:59.270 We're trying to solve for Nazrith today. 00:02:59.270 --> 00:03:01.450 So, let's say this is Nazrith. 00:03:01.450 --> 00:03:02.454 We'll say Naz for short. 00:03:06.100 --> 00:03:07.040 And let's say this is Omama. 00:03:11.320 --> 00:03:13.352 And this is today. 00:03:13.352 --> 00:03:16.780 I'm making a little chart. 00:03:16.780 --> 00:03:19.960 And let's say 4 years from now. 00:03:19.960 --> 00:03:21.580 4 -- oh, no, no. 00:03:21.580 --> 00:03:22.600 Not 4 years from now. 00:03:22.600 --> 00:03:24.640 I'm confusing myself. 00:03:24.640 --> 00:03:27.990 This would be 2 years ago. 00:03:27.990 --> 00:03:29.830 I'm still remembering the last problem. 00:03:29.830 --> 00:03:31.600 2 years ago. 00:03:31.600 --> 00:03:34.980 So this is today, 2 years into the future. 00:03:34.980 --> 00:03:37.980 Let me make a little chart. 00:03:37.980 --> 00:03:42.910 It's not the cleanest chart, but I think it'll do the job. 00:03:42.910 --> 00:03:44.500 So we're trying to solve for x. 00:03:44.500 --> 00:03:45.600 How old is Nazrith today. 00:03:45.600 --> 00:03:51.340 So let's just say Naz today that is equal to x. 00:03:51.340 --> 00:03:55.690 It says Nazrith is 4 years older than Omama. 00:03:55.690 --> 00:03:59.420 So this piece of information up here-- let me underline it. 00:03:59.420 --> 00:04:03.240 Nazrith is 4 years older Omama. 00:04:03.240 --> 00:04:07.530 Well, that piece of information tells us that Omama today, if 00:04:07.530 --> 00:04:12.890 Nazrith is x, then Omama's going to be x minus 4. 00:04:12.890 --> 00:04:17.740 Which makes sense because Naz is 4 years older than Omama. 00:04:17.740 --> 00:04:20.090 Now we go into the past. 00:04:20.090 --> 00:04:23.430 Let's write like this. 00:04:23.430 --> 00:04:27.680 So, 2 years ago -- so if Naz is x today, how old 00:04:27.680 --> 00:04:28.900 was she 2 years ago? 00:04:28.900 --> 00:04:32.050 Well, she was going to be x minus 2. 00:04:32.050 --> 00:04:33.080 That makes sense. 00:04:33.080 --> 00:04:34.940 If I'm 10 years old today, 2 years ago, 00:04:34.940 --> 00:04:35.970 I'll be 2 years less. 00:04:35.970 --> 00:04:36.960 I'd be 8 years old. 00:04:36.960 --> 00:04:40.900 So, if she's x today, 2 years ago she was x minus 2. 00:04:40.900 --> 00:04:44.690 And if Omama is x minus 4 today, than 2 years ago 00:04:44.690 --> 00:04:46.580 she'll be 2 less than that. 00:04:46.580 --> 00:04:49.730 So she'll be x minus 4 minus 2. 00:04:49.730 --> 00:04:53.230 And that equals x minus 6. 00:04:53.230 --> 00:04:53.940 Which makes sense. 00:04:53.940 --> 00:04:57.230 And, notice that she's still 4 years younger 00:04:57.230 --> 00:05:00.050 than Naz, or Nazrith. 00:05:00.050 --> 00:05:03.890 Now we have one final piece of information in this problem. 00:05:03.890 --> 00:05:12.330 It says, 2 years ago Nazrith was 5 times as old as Omama. 00:05:12.330 --> 00:05:14.980 So, we're in this situation. 00:05:14.980 --> 00:05:17.500 2 years ago -- that's what the sentence says, 2, years ago, so 00:05:17.500 --> 00:05:21.570 it's this situation -- Nazrith was 5 times as old as Omama. 00:05:21.570 --> 00:05:25.450 So Nazrith, which is x minus 2, was 5 times older than 00:05:25.450 --> 00:05:28.010 Omama, which is x minus 6. 00:05:28.010 --> 00:05:30.030 So let's just write that down. 00:05:30.030 --> 00:05:33.160 So, Nazrith, 2 years ago, was x minus 2. 00:05:33.160 --> 00:05:34.160 We get that from the chart. 00:05:34.160 --> 00:05:37.050 Nazrith was x minus 2, 2 years ago. 00:05:37.050 --> 00:05:41.140 And that sentence tells us, Nazrith was 5 times older 00:05:41.140 --> 00:05:42.670 than Omama was 2 years ago. 00:05:42.670 --> 00:05:44.860 Omama, 2 years ago, was x minus 6. 00:05:47.780 --> 00:05:49.670 Well, now we just solve for x. 00:05:49.670 --> 00:05:56.590 x minus 2 is equal to 5x minus 30. 00:05:56.590 --> 00:05:59.450 Now we could subtract x from both sides. 00:05:59.450 --> 00:06:02.280 And if you get 4x, I'm going to switch it around, to 00:06:02.280 --> 00:06:04.120 hopefully confuse you more. 00:06:04.120 --> 00:06:07.870 4x minus 30 equals minus 2. 00:06:07.870 --> 00:06:15.060 And then we get 4x is equal to 28 and x is equal to 7. 00:06:15.060 --> 00:06:16.880 So we solve the problem. 00:06:16.880 --> 00:06:20.160 Nazrith, today is 7 years old. 00:06:20.160 --> 00:06:21.560 And does this make sense? 00:06:21.560 --> 00:06:26.620 Well, if Nazrith today is 7 years old, then Omama today 00:06:26.620 --> 00:06:28.390 is going to be 3 years old. 00:06:28.390 --> 00:06:30.530 And that makes sense, because there's a 4 year difference. 00:06:30.530 --> 00:06:34.500 2 years ago, Nazrith was 5 years old. 00:06:34.500 --> 00:06:38.940 And 2 years ago Omama was only 1 year old. 00:06:38.940 --> 00:06:40.890 Because x minus 6. 00:06:40.890 --> 00:06:42.080 So it makes sense. 00:06:42.080 --> 00:06:44.570 Nazrith is 4 years older than Omama; Nazrith is 00:06:44.570 --> 00:06:47.430 7, Omama is 3 right now. 00:06:47.430 --> 00:06:50.130 2 years ago, when Nazrith was 5 times as old as Omama. 00:06:50.130 --> 00:06:53.340 2 years ago, Nazrith was 5 and Omama was 1. 00:06:53.340 --> 00:06:54.600 So it all works out. 00:06:54.600 --> 00:06:56.000 Pretty deep, no? 00:06:56.000 --> 00:06:57.990 Well, anyway, I think you're ready to at least try some 00:06:57.990 --> 00:06:59.200 of these age problems. 00:06:59.200 --> 00:07:01.120 And I'm going to do some more example problems. 00:07:01.120 --> 00:07:05.060 So that you can come back to this video page and see 00:07:05.060 --> 00:07:07.620 more if you're still a little bit confused. 00:07:07.620 --> 00:07:08.230 Have fun.

Algebra: Equation of a line

https://www.youtube.com/watch?v=gvwKv6F69F0

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https://www.youtube.com/api/timedtext?v=gvwKv6F69F0&ei=g2eUZanHKImfp-oPsYqpSA&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=5F0E38E07A186814E97EBC71AED1067822830FD0.568633CF6FDBBAFDCE2F9C5161BDE5D52D13BA9C&key=yt8&lang=en&fmt=vtt

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WEBVTT Kind: captions Language: en 00:00:01.110 --> 00:00:04.540 Welcome to the presentation on finding the equation of a line. 00:00:04.540 --> 00:00:05.410 Let's get started. 00:00:05.410 --> 00:00:06.345 Say I had two points. 00:00:06.345 --> 00:00:12.360 Let's say I have the point 1 comma 2, and I have the point 3 00:00:12.360 --> 00:00:14.360 comma 4, and I want to figure out the equation of the 00:00:14.360 --> 00:00:15.360 line through these points. 00:00:15.360 --> 00:00:19.640 So let's at least figure out what that line looks like. 00:00:19.640 --> 00:00:32.490 So 1 comma 2 is here, and 2, 3; 3, 4. 00:00:32.490 --> 00:00:39.130 3 comma 4 is here, and if I want to draw a line 00:00:39.130 --> 00:00:44.050 through them, it'll look something like that. 00:00:44.050 --> 00:00:46.375 So what we want to do is figure out the equation of this line. 00:00:46.375 --> 00:00:53.440 Well, we know the form of an equation of a line is y equals 00:00:53.440 --> 00:01:00.840 mx plus b, where m is the slope, and that tells you how 00:01:00.840 --> 00:01:03.620 steep the line is, and b is the y-intercept. 00:01:03.620 --> 00:01:05.430 And I don't know why people chose m and b. 00:01:05.430 --> 00:01:06.980 We'll have to do some research on that. 00:01:06.980 --> 00:01:09.270 b is the y-intercept and the y-intercept is just where does 00:01:09.270 --> 00:01:10.250 it intersect the y-axis. 00:01:13.250 --> 00:01:15.050 And this problem, you could actually look at it and 00:01:15.050 --> 00:01:17.940 figure it out, but let's do it mathematically. 00:01:17.940 --> 00:01:24.390 So the equation for the slope m: it's rise over run. 00:01:24.390 --> 00:01:27.640 Another way to view that is for any amount that you 00:01:27.640 --> 00:01:31.680 run along the x-axis, how much do you rise? 00:01:31.680 --> 00:01:33.350 Well, let's do that numerically. 00:01:33.350 --> 00:01:36.430 Rise is the same thing as change over y, and run is the 00:01:36.430 --> 00:01:37.800 same thing as change over x. 00:01:37.800 --> 00:01:41.570 Delta, this triangle, means change, change in y. 00:01:41.570 --> 00:01:44.650 Well, change in y, let's take the starting 00:01:44.650 --> 00:01:46.280 point to be 3 comma 4. 00:01:46.280 --> 00:01:48.980 Let's say we're going from 3 comma 4 to 2 comma 1. 00:01:48.980 --> 00:01:52.080 The change in y is 4 minus 2. 00:01:52.080 --> 00:02:01.830 We just took this 4 minus this 2 over 3 minus 1. 00:02:01.830 --> 00:02:02.550 My phone was ringing. 00:02:07.420 --> 00:02:10.760 And that's just this 3 minus this 1. 00:02:10.760 --> 00:02:16.560 So if we just solve for it, we get 4 minus 2 is 2, and 3 00:02:16.560 --> 00:02:21.280 minus 1 is also 2, so we get the slope is equal to 1. 00:02:21.280 --> 00:02:24.280 And that makes sense because when we move over 1 in x, 00:02:24.280 --> 00:02:26.270 we go up exactly 1 in y. 00:02:26.270 --> 00:02:32.790 When we move to the left 1 in x, we move down exactly 1 in y. 00:02:32.790 --> 00:02:38.490 So now we know the equation is y equals 1x plus b because 00:02:38.490 --> 00:02:40.880 we solved the m equals 1. 00:02:40.880 --> 00:02:42.580 And this is, of course, the same thing as 00:02:42.580 --> 00:02:46.200 y equals x plus b. 00:02:46.200 --> 00:02:48.760 Now, all we have left to do is solve for b. 00:02:48.760 --> 00:02:52.550 Well, how do we do that because we have three variables here. 00:02:52.550 --> 00:02:55.460 Well, we could actually substitute one of these pairs 00:02:55.460 --> 00:02:59.780 of points in for y and x, and that makes sense, because 00:02:59.780 --> 00:03:02.510 these points have to satisfy this equation. 00:03:02.510 --> 00:03:06.600 So let's take this first pair. y is equal to 2. 00:03:06.600 --> 00:03:10.360 2 is equal to x, which is 1 plus b. 00:03:10.360 --> 00:03:12.160 It's a pretty easy equation to solve. 00:03:12.160 --> 00:03:16.180 We get b equals 1, so that tells us that the equation of 00:03:16.180 --> 00:03:20.070 this line is y equals x plus 1. 00:03:20.070 --> 00:03:22.490 That's a pretty straightforward equation, and it makes sense. 00:03:22.490 --> 00:03:25.360 The y-intercept is 1, which is exactly here, 0 comma 00:03:25.360 --> 00:03:29.100 1, and the slope is 1, and that's pretty obvious. 00:03:29.100 --> 00:03:31.690 For every amount that we move to the right, we move the same 00:03:31.690 --> 00:03:34.150 amount up, so the slope is 1. 00:03:34.150 --> 00:03:35.110 Let's do another problem. 00:03:38.090 --> 00:03:40.200 Let's say I wanted to find the equation of the line between 00:03:40.200 --> 00:03:49.350 the points negative 3 comma 5 and 2 comma negative 6. 00:03:49.350 --> 00:03:51.260 Well, we do the same thing. 00:03:51.260 --> 00:03:55.690 m is equal to change in y over change in x. 00:03:55.690 --> 00:03:58.630 So let's take this as the starting point. 00:03:58.630 --> 00:04:01.955 So say negative 6 minus 5. 00:04:01.955 --> 00:04:08.300 So we just took negative 6 minus 5 over 2 00:04:08.300 --> 00:04:11.840 minus negative 3. 00:04:11.840 --> 00:04:13.650 You've got to be real careful to get the signs right. 00:04:13.650 --> 00:04:16.840 So it's 2 minus negative 3. 00:04:16.840 --> 00:04:22.485 Negative 6 minus 5 is minus 11, and 2 minus negative 3, well, 00:04:22.485 --> 00:04:26.350 that's the same thing as two plus plus 3, so that's 5. 00:04:26.350 --> 00:04:30.370 So we have the slope is equal to negative 11/5. 00:04:30.370 --> 00:04:35.450 And notice that if on the numerator we use negative 6 as 00:04:35.450 --> 00:04:37.630 the starting point, that in the denominator, we have to use 00:04:37.630 --> 00:04:39.940 2 as the starting point. 00:04:39.940 --> 00:04:41.090 We could have done it the other way around. 00:04:41.090 --> 00:04:48.240 We could have said 5 minus negative 6 over negative 3 00:04:48.240 --> 00:04:51.070 minus 2, in which case we would have gotten-- this would have 00:04:51.070 --> 00:04:54.570 been 11 over negative 5. 00:04:54.570 --> 00:04:58.305 So as long as you-- if you use the negative 6 first, then you 00:04:58.305 --> 00:05:00.840 have to use the 2 first, or if you use the 5 first, then 00:05:00.840 --> 00:05:01.885 you have to use the negative 3 first. 00:05:01.885 --> 00:05:04.080 I hope I'm not completely confusing you guys. 00:05:04.080 --> 00:05:07.500 Well, anyway, we know the slope is negative 11/5, so the 00:05:07.500 --> 00:05:15.680 equation of this line so far is y equals minus 11/5x plus b. 00:05:15.680 --> 00:05:18.410 Now we can take one of these pairs on the top and substitute 00:05:18.410 --> 00:05:20.090 back and solve for b. 00:05:20.090 --> 00:05:21.760 Let's take the first pair. 00:05:21.760 --> 00:05:23.390 So 5 is y. 00:05:23.390 --> 00:05:33.320 So we say 5 equals negative 3, so it's negative 11/5 00:05:33.320 --> 00:05:35.750 times negative 3, right? 00:05:35.750 --> 00:05:40.220 I just put the x in for x plus b. 00:05:40.220 --> 00:05:48.450 So just simplifying that, I get 5 is equal to 33/5 plus b, or b 00:05:48.450 --> 00:06:01.840 is equal to 5 minus 33/5, and this equals 25 minus 33/5. 00:06:01.840 --> 00:06:05.460 25 minus 33 is minus 8/5. 00:06:07.960 --> 00:06:10.580 So the equation of this line, and this one wasn't as clean as 00:06:10.580 --> 00:06:13.340 the other one, obviously, is-- let me do it in another color 00:06:13.340 --> 00:06:23.800 for emphasis-- y equals minus 11/5x minus 8/5. 00:06:26.400 --> 00:06:29.530 Hopefully, those two examples will give you enough of an idea 00:06:29.530 --> 00:06:34.570 to do the figuring out the equation of a line problems. 00:06:34.570 --> 00:06:36.820 And if you have problems with this, you might just want to 00:06:36.820 --> 00:06:39.750 try just the slope of the line problems or the y-intercept 00:06:39.750 --> 00:06:41.170 problems separately. 00:06:41.170 --> 00:06:43.160 I hope you have fun. 00:06:43.160 --> 00:06:44.460 Bye.

Level 3 exponents

https://www.youtube.com/watch?v=aYE26a5E1iU

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WEBVTT Kind: captions Language: en 00:00:01.120 --> 00:00:03.160 Welcome to level three exponents. 00:00:03.160 --> 00:00:05.200 Let's get started. 00:00:05.200 --> 00:00:12.590 So if I asked you what 4 to the 1/2 power is, your immediate 00:00:12.590 --> 00:00:14.660 inclination is to view this probably as like a 00:00:14.660 --> 00:00:18.800 multiplication problem and try to multiply it somehow or add 00:00:18.800 --> 00:00:19.750 them or something. 00:00:19.750 --> 00:00:21.430 And you always have to remind yourself this 00:00:21.430 --> 00:00:22.340 is not multiplication. 00:00:22.340 --> 00:00:24.730 I know when I first learned it, I always was tempted to do 00:00:24.730 --> 00:00:26.620 something with multiplication. 00:00:26.620 --> 00:00:29.820 Well, something to the 1/2 power might not be intuitive to 00:00:29.820 --> 00:00:32.870 you, but it actually turns out that this means the same thing 00:00:32.870 --> 00:00:36.260 as the square root of 4. 00:00:36.260 --> 00:00:40.390 Or another way, what times itself is equal to 4, 00:00:40.390 --> 00:00:43.600 and we know that the square root of 4 is 2. 00:00:43.600 --> 00:00:47.090 It could actually be a positive 2 or a negative 2 because we 00:00:47.090 --> 00:00:49.830 know that either of those numbers when they're 00:00:49.830 --> 00:00:51.530 squared could equal 4. 00:00:51.530 --> 00:00:54.090 But for the sake of this one we'll assume it's always 00:00:54.090 --> 00:00:55.110 the positive square root. 00:00:55.110 --> 00:00:58.610 So 4 to the 1/2 is equal to 2. 00:00:58.610 --> 00:01:05.510 Similarly, 9 to the 1/2, well that would be 3. 00:01:05.510 --> 00:01:09.710 16 to the 1/2 -- oops. 00:01:09.710 --> 00:01:13.270 16 to the 1/2, my subconscious gave away the answer. 00:01:13.270 --> 00:01:15.920 16 to the 1/2 power is 4. 00:01:15.920 --> 00:01:22.140 25 to the 1/2 power is 5. 00:01:22.140 --> 00:01:24.170 I think that might make sense to you now. 00:01:24.170 --> 00:01:27.740 So what does it mean when something is to the 1/3 power? 00:01:27.740 --> 00:01:32.950 Well, if I say 8 to the 1/3 power, you might 00:01:32.950 --> 00:01:34.260 already catch on. 00:01:34.260 --> 00:01:36.522 In the 1/2 power we said something times something 00:01:36.522 --> 00:01:38.190 is equal to 4. 00:01:38.190 --> 00:01:41.600 Well in the 1/3 power we have to say that something to the 00:01:41.600 --> 00:01:43.530 third power is equal to 8. 00:01:43.530 --> 00:01:46.730 And if you've been practicing your exponents you know that 2 00:01:46.730 --> 00:01:50.060 to the third power is equal to 8. 00:01:50.060 --> 00:01:55.060 So we know that 8 to the 1/3 is equal to 2. 00:01:55.060 --> 00:02:02.906 Similarly, 27 to the 1/3 is equal to 3. 00:02:02.906 --> 00:02:08.750 And 64 to the 1/3 is equal to 4. 00:02:08.750 --> 00:02:10.920 You might notice that I'm picking particular numbers, 00:02:10.920 --> 00:02:13.050 like 8 and 27, 64. 00:02:13.050 --> 00:02:15.060 That's because they have clean, cube roots. 00:02:15.060 --> 00:02:17.050 And then by the way, when something is to the 1/3 power 00:02:17.050 --> 00:02:20.480 that's the same thing as saying the cube root. 00:02:20.480 --> 00:02:22.190 I just used terminology without explaining it, 00:02:22.190 --> 00:02:25.250 which is very bad. 00:02:25.250 --> 00:02:28.900 So I just used 8 and 27 and 64 because when I raised them to 00:02:28.900 --> 00:02:31.010 the fractional exponents, they actually come out to 00:02:31.010 --> 00:02:31.740 be clean numbers. 00:02:31.740 --> 00:02:34.290 You could use a calculator and do something like 5 to the 1/3 00:02:34.290 --> 00:02:36.710 power and you'll get some weird decimal. 00:02:36.710 --> 00:02:37.695 Let's do some more problems. 00:02:41.980 --> 00:02:49.530 So we know that 9 to the 1/2 is equal to 3. 00:02:49.530 --> 00:02:54.930 Well what do you think 9 to the 2/3 is equal to? 00:02:54.930 --> 00:03:00.970 Well, it turns out that this is equivalent to 9 to the -- oops, 00:03:00.970 --> 00:03:05.580 I actually didn't want to do those -- what do you think 9 00:03:05.580 --> 00:03:08.860 to the 3/2 is equivalent to? 00:03:08.860 --> 00:03:14.650 Well this is the same thing as 9 to the 1/2 power 00:03:14.650 --> 00:03:15.580 to the third power. 00:03:15.580 --> 00:03:20.400 And I'll do a whole presentation on the actual 00:03:20.400 --> 00:03:24.020 principles of exponents, but it actually turns out you 00:03:24.020 --> 00:03:24.800 could just multiply. 00:03:24.800 --> 00:03:26.920 When you have one exponent to another is when you can 00:03:26.920 --> 00:03:28.920 multiply the two and that's where you get 3/2 . 00:03:28.920 --> 00:03:32.950 But 9 to the 1/2 we know is 3. 00:03:32.950 --> 00:03:36.690 And you're raising it to the third power, so that equals 27. 00:03:36.690 --> 00:03:38.740 I'm sure at this point I have confused you. 00:03:38.740 --> 00:03:42.050 Let's do more of these. 00:03:42.050 --> 00:03:47.370 So you know at this point that 16 to the 1/4 power -- 00:03:47.370 --> 00:03:48.270 think about what that is. 00:03:48.270 --> 00:03:51.500 That means that some number to the fourth power is 16. 00:03:51.500 --> 00:03:54.030 If you've been practicing your level one exponents, you'll 00:03:54.030 --> 00:03:58.820 probably know that well, that equals 2, because 2 times 2 00:03:58.820 --> 00:04:01.920 times 2 times 2, well that equals 16. 00:04:01.920 --> 00:04:06.320 So we know that 16 to the 1/4 is equal to 2. 00:04:06.320 --> 00:04:11.740 So what do you think 16 to the 2/4 is equal to? 00:04:11.740 --> 00:04:14.270 Well, we already know from that last problem that that's the 00:04:14.270 --> 00:04:21.710 same thing as 16 to the 1/4 squared -- that's the 2 on both 00:04:21.710 --> 00:04:26.340 sides -- and we know 16 to the 1/4 is 2, so that equals 2 00:04:26.340 --> 00:04:29.600 squared and that equals 4. 00:04:29.600 --> 00:04:32.790 And it all works out because we know from fractions another way 00:04:32.790 --> 00:04:35.200 to write the fraction 2/4 is to write 1/2. 00:04:35.200 --> 00:04:40.270 So this is the same thing as 16 to the 1/2 power. 00:04:40.270 --> 00:04:45.410 16 to the 1/2 power, well that's just equal to 4. 00:04:45.410 --> 00:04:48.790 Now I'm going to mix it up real good and do some negative 00:04:48.790 --> 00:04:50.790 fractional exponents. 00:04:50.790 --> 00:04:56.680 So what if I were to tell you 16 to the negative 1/2 power? 00:04:56.680 --> 00:04:59.400 Well this might seem very daunting at first, but as we 00:04:59.400 --> 00:05:01.460 know with the negative exponents level three, 00:05:01.460 --> 00:05:04.170 immediately we just say well this is the same thing as 1 00:05:04.170 --> 00:05:11.170 over 16 to the positive 1/2. 00:05:11.170 --> 00:05:17.910 And that's the same thing as 1 to the 1/2 over 16 to the 1/2. 00:05:17.910 --> 00:05:20.640 Well the square root of 1 is easy, it's 1. 00:05:20.640 --> 00:05:23.130 And 16 to the 1/2 is 4. 00:05:23.130 --> 00:05:24.300 So that wasn't too bad. 00:05:24.300 --> 00:05:26.500 It's a little daunting when you see a negative exponent, but 00:05:26.500 --> 00:05:29.640 immediately when you see that negative, just flip the 16 and 00:05:29.640 --> 00:05:33.320 then work it out like a regular fractional exponent problem. 00:05:33.320 --> 00:05:34.070 Let's do another one. 00:05:38.480 --> 00:05:51.980 8 over 27 to the negative 1/3. 00:05:51.980 --> 00:05:53.660 Immediately when we see that negative, we 00:05:53.660 --> 00:05:54.560 want to just flip it. 00:05:54.560 --> 00:06:03.440 So we'll say that equals 27 over 8 to the 1/3, and that 00:06:03.440 --> 00:06:11.060 equals 27 to the 1/3 over 8 to the 1/3. 00:06:11.060 --> 00:06:15.860 And we know that 27 to the 1/3, well that equals 3. 00:06:15.860 --> 00:06:20.720 And 8 to the 1/3, well that equals 2. 00:06:20.720 --> 00:06:26.275 So we've got 8 over 27 to the negative 1/3 is 3/2. 00:06:26.275 --> 00:06:28.170 The first problem probably looked very intimidating to 00:06:28.170 --> 00:06:30.280 you, but it only took us two steps to get there and as you 00:06:30.280 --> 00:06:32.330 do more practice, hopefully it'll seem more and 00:06:32.330 --> 00:06:34.070 more intuitive to you. 00:06:34.070 --> 00:06:36.760 Let me give you another problem. 00:06:36.760 --> 00:06:40.890 What's negative 8 to the negative third power? 00:06:43.990 --> 00:06:44.710 Let me change that. 00:06:44.710 --> 00:06:51.250 What's negative 8 to the negative 1/3 power? 00:06:51.250 --> 00:06:53.430 Once again, at first this might confuse you, but when you see 00:06:53.430 --> 00:06:55.590 that negative in the exponent, we just take the reciprocal of 00:06:55.590 --> 00:06:59.690 the base, so that we say that that is equal to negative 1 00:06:59.690 --> 00:07:06.680 over 8 to the 1/3 power. 00:07:06.680 --> 00:07:13.650 And we say well that is equal to, we could write it as 1 00:07:13.650 --> 00:07:16.770 over negative 8 to the 1/3. 00:07:16.770 --> 00:07:21.280 We say what number times itself 3 times is equal to negative 8? 00:07:21.280 --> 00:07:24.540 Well, we know from intuition there's no real mechanical way 00:07:24.540 --> 00:07:28.440 to do this, but we know that negative 2 times negative 2 is 00:07:28.440 --> 00:07:31.060 4 times negative 2 is negative 8. 00:07:31.060 --> 00:07:34.340 So we know that this is equal to 1 over negative 00:07:34.340 --> 00:07:37.956 2 or negative 1/2. 00:07:37.956 --> 00:07:41.175 So negative eight to the negative 1/3 is 00:07:41.175 --> 00:07:45.210 equal to minus 1/2. 00:07:45.210 --> 00:07:48.760 Let's do another one, one more problem just to thoroughly 00:07:48.760 --> 00:07:51.070 melt your brain. 00:07:51.070 --> 00:08:03.260 Let's say 9 over 4 to the negative 3 over 2. 00:08:03.260 --> 00:08:06.170 Well, immediately we see that negative exponent, 00:08:06.170 --> 00:08:07.670 let's flip the base. 00:08:07.670 --> 00:08:13.250 We get 4 over 9 to the 3/2 . 00:08:13.250 --> 00:08:22.510 Well we know that that equals 4 over 9 to the 1/2, and all 00:08:22.510 --> 00:08:25.700 of that to the third power. 00:08:25.700 --> 00:08:28.100 4 over 9 to the 1/2, I think at this point you know that's the 00:08:28.100 --> 00:08:31.470 same thing as 4 to the 1/2 which is 2 over 9 to 00:08:31.470 --> 00:08:33.400 the 1/2 which is 3. 00:08:33.400 --> 00:08:37.520 Now we have to just raise everything to the third power. 00:08:37.520 --> 00:08:39.950 And that's the same thing as 2 the third power which is 00:08:39.950 --> 00:08:45.400 8 divided by 3 to the third power. 00:08:45.400 --> 00:08:48.030 Well that's 27. 00:08:48.030 --> 00:08:52.470 There we have 9/4 to the negative 3/2 power is 00:08:52.470 --> 00:08:55.090 equal to 8 over 27. 00:08:55.090 --> 00:08:57.410 Now hopefully you can at least do these problems. 00:08:57.410 --> 00:09:00.920 You probably don't have a good intuitive sense for exactly 00:09:00.920 --> 00:09:04.650 what a negative 3/2 power is, and hopefully I can cover that 00:09:04.650 --> 00:09:06.580 for you in future modules. 00:09:06.580 --> 00:09:08.710 But I think you're ready to try some of the level three 00:09:08.710 --> 00:09:10.310 exponent practice problems. 00:09:10.310 --> 00:09:11.830 Have fun.

Level 1 multiplying expressions

https://www.youtube.com/watch?v=Sc0e6xrRJYY

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WEBVTT Kind: captions Language: en 00:00:01.100 --> 00:00:03.570 Welcome to the presentation on multiplying expressions. 00:00:03.570 --> 00:00:05.260 Let's get started. 00:00:05.260 --> 00:00:11.540 So if I were to ask you what x plus 2 times, let me use a 00:00:11.540 --> 00:00:20.610 different color, times x plus 3 equals, at first you'd be like, 00:00:20.610 --> 00:00:22.450 huh, that's a little strange. 00:00:22.450 --> 00:00:24.180 But it turns out that you actually already know 00:00:24.180 --> 00:00:24.940 how to do this problem. 00:00:24.940 --> 00:00:27.060 And it's just the distributive property. 00:00:27.060 --> 00:00:30.040 Because, if I just had -- let me write a different problem 00:00:30.040 --> 00:00:33.603 here -- if I just wrote a let me just make it clear, this is 00:00:33.603 --> 00:00:41.400 a separate problem -- if I just wrote a times x plus 3, you 00:00:41.400 --> 00:00:58.060 know that that's just a x plus, you could say a 3, or 00:00:58.060 --> 00:01:00.510 another way to say that would be 3a, right? 00:01:00.510 --> 00:01:04.020 But as you see, all we did is distribute it, this a times 3. 00:01:04.020 --> 00:01:07.290 All we did is distribute the a times the x and the 3. 00:01:07.290 --> 00:01:08.763 Well, we're going to do the same thing here, but instead 00:01:08.763 --> 00:01:10.630 of a we have x plus 2. 00:01:10.630 --> 00:01:12.050 So let's do that. 00:01:12.050 --> 00:01:18.830 So this would be -- let me switch to green -- we get x 00:01:18.830 --> 00:01:22.995 plus 2, and since I'm using green, I'm just going 00:01:22.995 --> 00:01:24.482 to stay with green. 00:01:24.482 --> 00:01:28.400 x plus 2, let me switch to orange. 00:01:28.400 --> 00:01:33.980 x plus 2 times x plus x plus 2 times 3. 00:01:33.980 --> 00:01:35.310 Does that make sense? 00:01:35.310 --> 00:01:36.720 If that looks a little confusing, just pretend 00:01:36.720 --> 00:01:39.550 like that x plus 2 is the a in this example. 00:01:39.550 --> 00:01:42.080 And all we did is, we distributed it across 00:01:42.080 --> 00:01:44.110 that x plus three. 00:01:44.110 --> 00:01:46.980 And now it becomes a pretty straightforward problem. 00:01:46.980 --> 00:01:50.140 What is -- we do distribution again with each part 00:01:50.140 --> 00:01:51.420 of this problem. 00:01:51.420 --> 00:01:54.740 So, let me stay in orange. 00:01:54.740 --> 00:01:56.950 What is x times x? 00:01:56.950 --> 00:02:00.820 Well that's x squared. 00:02:00.820 --> 00:02:04.850 And then x times 2 -- well, that's 2x. 00:02:04.850 --> 00:02:06.860 So we did this left-hand side already. 00:02:06.860 --> 00:02:08.680 Now we do this right-hand side. 00:02:08.680 --> 00:02:10.730 What's 3 times x? 00:02:10.730 --> 00:02:13.650 Well that's 3x. 00:02:13.650 --> 00:02:14.940 What's 3 times 2? 00:02:14.940 --> 00:02:17.680 Well, that's 6. 00:02:17.680 --> 00:02:19.680 And now we're almost done. 00:02:19.680 --> 00:02:21.670 We could say, oh, we have a 2x here and a 3x here, we can 00:02:21.670 --> 00:02:23.700 simplify and add those together. 00:02:23.700 --> 00:02:26.070 And we know that that equals x squared. 00:02:26.070 --> 00:02:28.000 2x plus 3x is equal to 5x. 00:02:28.000 --> 00:02:32.930 x squared plus 5x plus 6. 00:02:32.930 --> 00:02:34.890 So all we did, there's really nothing new to learn here. 00:02:34.890 --> 00:02:39.460 We just distributed the x plus 2 times each term of x plus 3. 00:02:39.460 --> 00:02:40.630 And we got the second step. 00:02:40.630 --> 00:02:43.480 And then we distributed this x times x plus 2, and this 3 00:02:43.480 --> 00:02:45.660 times x plus 2, and simplify. 00:02:45.660 --> 00:02:46.810 Let's do a couple of problems. 00:02:46.810 --> 00:02:51.970 And hopefully it'll hit the point home. 00:02:51.970 --> 00:03:07.180 Let's say I had 5x plus 9 times 4x minus 2. 00:03:07.180 --> 00:03:16.590 Once again, that equals 5x plus 9. 00:03:16.590 --> 00:03:18.120 I'm just distributing it. 00:03:18.120 --> 00:03:20.640 I want to stay in my color that I'm using, so that's why I'm 00:03:20.640 --> 00:03:22.670 just writing it out like this. 00:03:22.670 --> 00:03:27.660 That equals 5x plus 9 times 4x. 00:03:27.660 --> 00:03:32.350 And then plus, 5x plus 9 times minus 2. 00:03:32.350 --> 00:03:36.570 That minus 2 is here and this 4x is here. 00:03:36.570 --> 00:03:37.830 And now we just multiply it out. 00:03:37.830 --> 00:03:45.170 5x times 4x is 20x squared. 00:03:45.170 --> 00:03:46.720 Hope that makes sense to you; let me write it 00:03:46.720 --> 00:03:47.756 down in this corner. 00:03:47.756 --> 00:03:55.050 5x times 4x is the same thing as 5 times 4 times x times x. 00:03:55.050 --> 00:03:57.040 That equals 20x squared. 00:03:57.040 --> 00:03:58.840 Hopefully that makes sense to you. 00:03:58.840 --> 00:04:00.290 So, going back to the problem. 00:04:00.290 --> 00:04:05.830 5x times 4x is 20x squared plus 9 times 4x is 36x. 00:04:08.760 --> 00:04:13.040 Plus minus 2 times 5x. 00:04:13.040 --> 00:04:17.000 So that's minus 10x. 00:04:17.000 --> 00:04:20.860 And then 9 times negative 2, well, that's minus 18. 00:04:20.860 --> 00:04:22.470 And we're almost done. 00:04:22.470 --> 00:04:24.696 So we get 20x squared. 00:04:24.696 --> 00:04:27.310 And we have 36x. 00:04:27.310 --> 00:04:28.990 Minus 10x. 00:04:28.990 --> 00:04:34.040 So that's plus 26x minus 18. 00:04:34.040 --> 00:04:34.520 There. 00:04:34.520 --> 00:04:36.430 We're done. 00:04:36.430 --> 00:04:37.325 Let's do another problem. 00:04:42.110 --> 00:04:49.710 Let's do 2x plus y. 00:04:49.710 --> 00:04:52.640 Whoops, some parentheses. 00:04:52.640 --> 00:04:55.040 I'll stay in one color, since I think you understand 00:04:55.040 --> 00:04:56.170 what we're doing. 00:04:56.170 --> 00:05:03.210 Times 3x plus 2y. 00:05:03.210 --> 00:05:06.790 Well, once again, this is the same thing as -- and we could 00:05:06.790 --> 00:05:08.100 just do it in a little different way, a little 00:05:08.100 --> 00:05:08.840 different order. 00:05:08.840 --> 00:05:10.350 But we could distribute it like this. 00:05:10.350 --> 00:05:15.960 We could say that this is 3x times 2x plus y. 00:05:15.960 --> 00:05:21.570 Plus 2y times 2x plus y. 00:05:21.570 --> 00:05:22.050 See what I did? 00:05:22.050 --> 00:05:23.280 I just switched the order this time. 00:05:23.280 --> 00:05:24.410 Just to mix things up. 00:05:24.410 --> 00:05:26.060 This 3x is there. 00:05:26.060 --> 00:05:27.140 This 2y is here. 00:05:27.140 --> 00:05:31.320 And we just distributed it along 2 -- and we just, we 00:05:31.320 --> 00:05:35.260 distributed it 2x plus y along each of these numbers. 00:05:35.260 --> 00:05:36.540 And now we just multiply it out. 00:05:36.540 --> 00:05:40.630 3x times 2x is 6x squared. 00:05:40.630 --> 00:05:46.970 3x times y -- that's 3xy. 00:05:46.970 --> 00:05:50.930 3xy is just another way of saying 3 times x times y. 00:05:50.930 --> 00:05:56.850 Plus, now, 2 times y times 2y times 2x, well, that's 4 y x. 00:05:59.800 --> 00:06:01.320 4 y x. 00:06:01.320 --> 00:06:04.450 2 times 2 times y times x. 00:06:04.450 --> 00:06:08.020 Plus 2y times y. 00:06:08.020 --> 00:06:11.370 Well, that's just 2y squared. 00:06:11.370 --> 00:06:13.590 Now, can we simplify anything here? 00:06:13.590 --> 00:06:14.980 Think about it a little bit. 00:06:14.980 --> 00:06:17.960 Well, it turns out that x y and y are actually the same thing. 00:06:17.960 --> 00:06:19.570 They're just switching the order. 00:06:19.570 --> 00:06:24.300 So you could just rewrite this as 6x squared plus 00:06:24.300 --> 00:06:29.110 7 x y plus 2y squared. 00:06:29.110 --> 00:06:31.560 And we could have just as easily have written it as 00:06:31.560 --> 00:06:39.471 6x squared plus 7y x plus 2y squared. 00:06:39.471 --> 00:06:41.420 I hope I'm not confusing you. 00:06:41.420 --> 00:06:42.910 Let's do one more, real fast. 00:06:45.800 --> 00:06:53.100 Let's do 2x plus 2 squared. 00:06:53.100 --> 00:06:55.870 Well, you might be temptd to just square each of these 00:06:55.870 --> 00:06:57.180 terms, but you've got to be careful because you 00:06:57.180 --> 00:06:58.340 have this x here. 00:06:58.340 --> 00:07:01.840 So it actually turns out that this is equal to 2x 00:07:01.840 --> 00:07:06.230 plus 2 times 2x plus 2. 00:07:06.230 --> 00:07:08.740 Any number squared is just that number times itself. 00:07:08.740 --> 00:07:11.950 So any expression like 2x plus 2 squared, is just that 00:07:11.950 --> 00:07:14.010 expression times itself. 00:07:14.010 --> 00:07:16.980 And now we can just do it as -- this is the same thing 00:07:16.980 --> 00:07:22.310 as 2x times 2x plus 2. 00:07:22.310 --> 00:07:26.070 Plus 2 times 2x plus 2. 00:07:26.070 --> 00:07:27.330 We just multiply everything out. 00:07:27.330 --> 00:07:31.240 2x times 2x, that's 4x squared. 00:07:31.240 --> 00:07:35.530 2x times 2, that's 4x. 00:07:35.530 --> 00:07:39.690 Plus 2 times 2x is 4x again. 00:07:39.690 --> 00:07:41.802 And then 2 times 2, well, that's 4. 00:07:41.802 --> 00:07:43.770 And we're almost there. 00:07:43.770 --> 00:07:45.750 We just can add up these two terms. 00:07:45.750 --> 00:07:51.140 4x squared plus 8x plus 4. 00:07:51.140 --> 00:07:52.520 And we're done. 00:07:52.520 --> 00:07:55.900 I think you're ready now to try some level one multiplying 00:07:55.900 --> 00:07:57.220 expression problems. 00:07:57.220 --> 00:07:58.930 Hope you have fun.

Level 2 Exponents

https://www.youtube.com/watch?v=1Nt-t9YJM8k

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en

WEBVTT Kind: captions Language: en 00:00:00.720 --> 00:00:03.582 Welcome to the presentation on the Level 2 exponents. 00:00:03.582 --> 00:00:06.480 In Level 2 exponents, the only thing we're going to add to 00:00:06.480 --> 00:00:09.740 the mix now is the concept of a negative exponent. 00:00:09.740 --> 00:00:14.180 So we learned already that 2 the third power, well, that 00:00:14.180 --> 00:00:17.140 just equals 2 times 2 times 2. 00:00:17.140 --> 00:00:19.110 Hopefully, by now, that's second nature to you, 00:00:19.110 --> 00:00:21.320 and that equals 8. 00:00:21.320 --> 00:00:24.960 Now I'm going to teach you what 2 to the negative 00:00:24.960 --> 00:00:26.630 third power is. 00:00:26.630 --> 00:00:28.490 I know a lot of you are going to say, oh, no, that's negative 00:00:28.490 --> 00:00:31.100 8, but whenever you see exponents, you always have to 00:00:31.100 --> 00:00:34.260 remind yourself, exponents are not multiplication. 00:00:34.260 --> 00:00:36.650 I know there's that temptation to say, well, 2 times negative 00:00:36.650 --> 00:00:39.720 3 is negative 6, so maybe 2 to the negative third power 00:00:39.720 --> 00:00:42.300 is negative 8, but that's not the case. 00:00:42.300 --> 00:00:45.920 And I'll explain in a future module why we use this 00:00:45.920 --> 00:00:49.220 convention, but 2 to the negative third power, it 00:00:49.220 --> 00:00:57.580 turns out, is equivalent to 1/2 to the third power. 00:00:57.580 --> 00:01:00.280 So it turns out that the negative exponent, what it does 00:01:00.280 --> 00:01:03.440 is it means to take the inverse of the base-- we'll call the 00:01:03.440 --> 00:01:06.760 number 2 as the base-- and take that to the positive 00:01:06.760 --> 00:01:08.070 version of the exponent. 00:01:08.070 --> 00:01:10.020 And 1/2 to the third power? 00:01:10.020 --> 00:01:16.336 Well, we already learned that's 1/2 times 1/2 times 1/2, 00:01:16.336 --> 00:01:19.170 and that equals 1/8. 00:01:19.170 --> 00:01:22.600 So we say 2 to the negative third power. 00:01:22.600 --> 00:01:24.040 That didn't come out right. 00:01:24.040 --> 00:01:29.300 2 to the negative third power is equal to 1/8. 00:01:29.300 --> 00:01:31.810 Let's do another one. 00:01:31.810 --> 00:01:38.530 Let's say 3 to the negative 2 power. 00:01:38.530 --> 00:01:41.300 Once again, immediately when we see that negative in the 00:01:41.300 --> 00:01:43.420 exponent, the easiest thing to do is just immediately take 00:01:43.420 --> 00:01:44.882 the reciprocal of the base. 00:01:44.882 --> 00:01:46.780 So we take 1/3. 00:01:46.780 --> 00:01:49.910 And we raise that to the positive 2 power. 00:01:49.910 --> 00:01:51.040 And that's easy enough. 00:01:51.040 --> 00:01:55.261 1/3 squared, well, that's equal to 1/9. 00:01:55.261 --> 00:01:56.700 Let's do some more problems. 00:02:00.170 --> 00:02:12.270 What if I had 2/3 to the negative third power? 00:02:12.270 --> 00:02:14.960 Once again, just to make it simple, whenever I see that 00:02:14.960 --> 00:02:16.900 negative in the exponent, I want to get rid of it. 00:02:16.900 --> 00:02:20.100 So I immediately take the reciprocal of the base. 00:02:20.100 --> 00:02:25.430 The reciprocal of 2/3 is 3/2, and I raise that to the 00:02:25.430 --> 00:02:27.180 positive third power. 00:02:27.180 --> 00:02:29.495 So what changed between the left and the right side? 00:02:29.495 --> 00:02:33.300 The 2/3 I flipped, and I turned the negative 00:02:33.300 --> 00:02:34.530 3 into a positive 3. 00:02:34.530 --> 00:02:37.510 And now this just becomes a Level 1 exponent. 00:02:37.510 --> 00:02:52.710 This equals 3/2 times 3/2 times 3/2, and that equals 27/8. 00:02:52.710 --> 00:02:53.970 So that's interesting. 00:02:53.970 --> 00:02:58.150 2/3 to the negative 3 is equal to 27/8. 00:02:58.150 --> 00:02:59.090 Let's do some more. 00:03:02.740 --> 00:03:12.190 Let's do 4/7 to the negative 1. 00:03:12.190 --> 00:03:16.870 Once again, we have a negative number in the exponent. 00:03:16.870 --> 00:03:21.700 That's the same thing as taking the reciprocal of the base 00:03:21.700 --> 00:03:23.720 and raising it to the positive exponent. 00:03:23.720 --> 00:03:26.425 Well, 7/4 to the 1, any number to the first power 00:03:26.425 --> 00:03:27.570 is just the same number. 00:03:27.570 --> 00:03:29.380 It's equal to 7/4. 00:03:29.380 --> 00:03:31.770 So when take it to the negative 1 power, all you're essentially 00:03:31.770 --> 00:03:34.490 doing is getting the reciprocal of the number. 00:03:34.490 --> 00:03:37.440 Let's do some more problems. 00:03:37.440 --> 00:03:41.400 2 to the negative 5. 00:03:41.400 --> 00:03:46.120 Once again, we take the reciprocal of 2, and we say 00:03:46.120 --> 00:03:51.680 1/2, and now that can be raised to the fifth power, and that 00:03:51.680 --> 00:04:00.520 equals 1/2 times 1/2 times 1/2 times 1/2 times 1/2, 00:04:00.520 --> 00:04:04.750 and that equals 1/32. 00:04:04.750 --> 00:04:07.030 Another way we could have viewed 2 to the negative fifth 00:04:07.030 --> 00:04:12.040 is that, 2 to the negative fifth, we could have said that 00:04:12.040 --> 00:04:16.210 equals 1 over 2 to the fifth. 00:04:16.210 --> 00:04:19.000 And we know that 2 to the fifth is 32, so that would've 00:04:19.000 --> 00:04:20.380 been the same thing. 00:04:20.380 --> 00:04:23.380 Two ways to do it, pretty much just changing the order of when 00:04:23.380 --> 00:04:25.180 you flip versus when you actually calculate 00:04:25.180 --> 00:04:27.210 the exponent. 00:04:27.210 --> 00:04:31.110 Let me do two or three more problems. 00:04:31.110 --> 00:04:33.360 And after I write down each of these problems, you might just 00:04:33.360 --> 00:04:35.780 want to pause it and see if you can do the problem yourself, 00:04:35.780 --> 00:04:38.100 and then compare your answer to mine. 00:04:38.100 --> 00:04:46.760 So let's say I had negative 4 to the negative third power. 00:04:46.760 --> 00:04:49.090 Immediately, I like to get rid of the negative in the 00:04:49.090 --> 00:04:56.120 exponent, and I know that that equals minus 1/4 to the third 00:04:56.120 --> 00:05:03.500 power, and that equals minus 1/4 times minus 1/4 00:05:03.500 --> 00:05:06.110 times minus 1/4. 00:05:06.110 --> 00:05:08.630 The negative times a negative is a positive, but then we're 00:05:08.630 --> 00:05:11.810 multiplying that times another negative, so we get a negative. 00:05:11.810 --> 00:05:14.680 1 times 1 times 1 is 1. 00:05:14.680 --> 00:05:18.360 4 times 4 is 16 times 4 is 64. 00:05:18.360 --> 00:05:23.340 So it equals negative 1/64. 00:05:23.340 --> 00:05:24.340 Let's do another problem. 00:05:27.210 --> 00:05:29.100 Let me think of a good number. 00:05:29.100 --> 00:05:31.150 8/9. 00:05:31.150 --> 00:05:33.270 Let's make it negative. 00:05:33.270 --> 00:05:40.230 Negative 8/9 to the negative second power. 00:05:40.230 --> 00:05:44.750 Well, once again, that equals negative 9/8. 00:05:44.750 --> 00:05:48.560 Notice, I immediately just took the reciprocal of the base to 00:05:48.560 --> 00:05:54.630 the positive 2 power, and now that equals negative 9/8 00:05:54.630 --> 00:05:57.480 times negative 9/8. 00:05:57.480 --> 00:06:00.180 A negative times a negative is a positive, so we 00:06:00.180 --> 00:06:05.230 get 9 times 9 is 81/64. 00:06:05.230 --> 00:06:06.260 I think you get the point now. 00:06:06.260 --> 00:06:08.730 The only new thing we've learned, really, is that when 00:06:08.730 --> 00:06:12.790 you have a negative exponent, it's the same thing as taking 00:06:12.790 --> 00:06:14.710 the reciprocal of the base and raising it to the 00:06:14.710 --> 00:06:16.250 positive exponent. 00:06:16.250 --> 00:06:19.150 Hopefully, that last statement didn't confuse you more, did 00:06:19.150 --> 00:06:21.630 more good than damage, but I think you're ready to 00:06:21.630 --> 00:06:22.620 try some problems now. 00:06:22.620 --> 00:06:24.150 Have fun!

Solving a quadratic by factoring

https://www.youtube.com/watch?v=N30tN9158Kc

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WEBVTT Kind: captions Language: en 00:00:01.090 --> 00:00:03.050 Welcome to solving a quadratic by factoring. 00:00:03.050 --> 00:00:04.970 Let's start doing some problems. 00:00:04.970 --> 00:00:13.480 So, let's say I had a function f of x is equal to x 00:00:13.480 --> 00:00:18.410 squared plus 6x plus 8. 00:00:18.410 --> 00:00:23.810 Now if I were to graph f of x, the graph is going to 00:00:23.810 --> 00:00:24.790 look something like this. 00:00:24.790 --> 00:00:27.810 I don't know exactly what it's going to look like, but it's 00:00:27.810 --> 00:00:31.420 going to be a parabola and it's going to intersect the x-axis 00:00:31.420 --> 00:00:34.440 at a couple of points, here and here. 00:00:34.440 --> 00:00:36.460 And what we're going to try to do is determine what 00:00:36.460 --> 00:00:38.480 those two points are. 00:00:38.480 --> 00:00:40.800 So first of all, when a function intersects the 00:00:40.800 --> 00:00:44.220 x-axis, that means f of x is equal to zero. 00:00:44.220 --> 00:00:47.820 Because this is f of x-axis, similar to the y-axis. 00:00:47.820 --> 00:00:49.760 So here f of x is 0. 00:00:49.760 --> 00:00:52.420 So in order to solve this equation we set f of x to 0, 00:00:52.420 --> 00:01:01.480 and we get x squared plus 6x plus 8 is equal to 0. 00:01:01.480 --> 00:01:04.120 Now this might look like you could solve it pretty easily, 00:01:04.120 --> 00:01:07.060 but that x squared term messes things up and you could 00:01:07.060 --> 00:01:09.130 try it out for yourself. 00:01:09.130 --> 00:01:10.680 So we're going to do is factor this. 00:01:10.680 --> 00:01:13.660 And we're going to say that x squared plus 6x plus 8, but 00:01:13.660 --> 00:01:19.240 this can be written as x plus something times x 00:01:19.240 --> 00:01:20.890 plus something. 00:01:20.890 --> 00:01:23.200 It will still equal that, except that's equal to 0. 00:01:23.200 --> 00:01:25.650 Now in this presentation, I'm going to just show you the 00:01:25.650 --> 00:01:27.980 systematic or you could say the mechanical way of doing this. 00:01:27.980 --> 00:01:30.720 I'm going to give you another presentation on why this works. 00:01:30.720 --> 00:01:33.420 And you might want to just multiply out the answers we 00:01:33.420 --> 00:01:36.140 get in and multiply out the expressions and 00:01:36.140 --> 00:01:38.400 see why it works. 00:01:38.400 --> 00:01:41.160 And the message we're going to use is, we look at the 00:01:41.160 --> 00:01:43.405 coefficient on this x term, 6. 00:01:43.405 --> 00:01:46.450 And we say what two numbers will add up to 6. 00:01:46.450 --> 00:01:49.840 And when those same two numbers are multiplied 00:01:49.840 --> 00:01:51.650 you're going to get 8. 00:01:51.650 --> 00:01:53.290 Well let's just think about the factors of 8. 00:01:53.290 --> 00:02:00.160 The factors of 8 are one to 4 and 8. 00:02:00.160 --> 00:02:05.670 well 1 times 8 is 8, but 1 plus 8 is 9, so that doesn't work. 00:02:05.670 --> 00:02:09.610 2 times 4 is 8, and 2 plus 4 is 6. 00:02:09.610 --> 00:02:11.040 So that works. 00:02:11.040 --> 00:02:17.560 So we could just say x plus 2 and x plus 4 is equal to 0. 00:02:17.560 --> 00:02:21.150 Now if two expressions or two numbers times each other equals 00:02:21.150 --> 00:02:24.470 0, that means that one of those two numbers or both of 00:02:24.470 --> 00:02:25.990 them must equal 0. 00:02:25.990 --> 00:02:37.785 So now we can say that x plus 2 equals 0, and x 00:02:37.785 --> 00:02:41.880 plus 4 is equal to zero. 00:02:41.880 --> 00:02:43.750 Well, this is a very simple equation. 00:02:43.750 --> 00:02:47.680 We subtract 2 from both sides and we get x equals negative 2. 00:02:47.680 --> 00:02:53.030 And here we get x equals minus 4. 00:02:53.030 --> 00:02:55.020 And if we substitute either of these into the 00:02:55.020 --> 00:02:57.470 original equation, we'll see that it works. 00:02:57.470 --> 00:03:00.650 Minus 2-- so let's just try it with minus 2 and I'll leave 00:03:00.650 --> 00:03:07.060 minus 4 up to you --so minus 2 squared plus 6 times 00:03:07.060 --> 00:03:09.940 minus 2 plus 8. 00:03:09.940 --> 00:03:18.070 Minus 2 squared is 4, minus 12-- 6 times minus 2 --plus 8. 00:03:18.070 --> 00:03:21.170 And sure enough that equals 0. 00:03:21.170 --> 00:03:23.030 And if you did the same thing with negative 4, you'd 00:03:23.030 --> 00:03:24.810 also see that works. 00:03:24.810 --> 00:03:26.600 And you might be saying, wow, this is interesting. 00:03:26.600 --> 00:03:30.180 This is an equation that has two solutions. 00:03:30.180 --> 00:03:33.040 Well, if you think about it, it makes sense because the graph 00:03:33.040 --> 00:03:39.130 of f of x is intersecting the x-axis in two different places. 00:03:39.130 --> 00:03:40.150 Let's do another problem. 00:03:44.470 --> 00:03:56.580 Let's say I had f of x is equal to 2 x squared 00:03:56.580 --> 00:04:01.830 plus 20x plus 50. 00:04:01.830 --> 00:04:04.170 So if we want to figure out where it intersects the x-axis, 00:04:04.170 --> 00:04:07.470 we just set f of x equal to 0, and I'll just swap the left and 00:04:07.470 --> 00:04:09.665 right sides of the equation. 00:04:09.665 --> 00:04:20.520 And I get 2x squared plus 20x plus 50 equals 0. 00:04:20.520 --> 00:04:22.890 Now, what's a little different this time from last time, is 00:04:22.890 --> 00:04:25.910 here the coefficient on that x squared is actually a 2 instead 00:04:25.910 --> 00:04:28.140 of a 1, and I like it to be a 1. 00:04:28.140 --> 00:04:30.680 So let's divide the whole equation, both the left 00:04:30.680 --> 00:04:33.150 and right sides, by 2. 00:04:33.150 --> 00:04:42.460 I get x squared plus 10x plus 25 equals 0. 00:04:42.460 --> 00:04:45.786 So all I did is I multiplied 1/2 times-- this is the same 00:04:45.786 --> 00:04:48.220 thing as dividing by 2 --times 1/2. 00:04:48.220 --> 00:04:50.940 And of course 0 times 1/2 is 0. 00:04:50.940 --> 00:04:52.580 Now we are ready to do what we did before, and you 00:04:52.580 --> 00:04:54.660 might want to pause it and try it yourself. 00:04:54.660 --> 00:05:00.080 We're going to say x plus something times x plus 00:05:00.080 --> 00:05:05.450 something is equal to 0 and those two somethings, they 00:05:05.450 --> 00:05:08.300 should add up to 10, and when you multiply them, 00:05:08.300 --> 00:05:09.940 they should be 25. 00:05:09.940 --> 00:05:11.690 Let's think about the factors of 25. 00:05:11.690 --> 00:05:16.120 You have 1, 5, and 25. 00:05:16.120 --> 00:05:18.100 Well 1 times 25 is 25. 00:05:18.100 --> 00:05:22.260 1 plus 25 is 26, not 10. 00:05:22.260 --> 00:05:30.170 5 times 5 is 25, and 5 plus 5 is 10, so 5 actually works. 00:05:30.170 --> 00:05:34.640 It actually turns out that both of these numbers are 5. 00:05:34.640 --> 00:05:39.690 So you get x plus 5 equals 0 or x plus 5 equals 0. 00:05:39.690 --> 00:05:42.882 So you just have to really write it once. 00:05:42.882 --> 00:05:45.590 So you get x equals negative 5. 00:05:45.590 --> 00:05:47.030 So how do you think about this graphically? 00:05:47.030 --> 00:05:50.290 I just told you that these equations can intersect the 00:05:50.290 --> 00:05:53.470 x-axis in two places, but this only has one solution. 00:05:53.470 --> 00:05:54.920 Well, this solution would look like. 00:05:58.960 --> 00:06:05.080 If this is x equals negative 5, we'd have a parabola that just 00:06:05.080 --> 00:06:08.020 touches right there, and then comes back up. 00:06:08.020 --> 00:06:09.880 And instead of intersecting in two places it only 00:06:09.880 --> 00:06:13.580 intersects right there at x equals negative 5. 00:06:13.580 --> 00:06:17.110 And now as an exercise just to prove to you that I'm not 00:06:17.110 --> 00:06:24.280 teaching you incorrectly, let's multiply x plus 5 times x plus 00:06:24.280 --> 00:06:29.370 5 just to show you that it equals what it should equal. 00:06:29.370 --> 00:06:32.730 So we just say that this is the same thing is x times x plus 00:06:32.730 --> 00:06:39.950 5 plus 5 times x plus 5. 00:06:39.950 --> 00:06:47.200 x squared plus 5x plus 5x plus 25. 00:06:47.200 --> 00:06:52.410 And that's x squared plus 10x plus 25. 00:06:52.410 --> 00:06:54.780 So, it equals what we said it should equal. 00:06:54.780 --> 00:06:57.320 And I'm going to once again do another module where I explain 00:06:57.320 --> 00:06:59.870 this a little bit more. 00:06:59.870 --> 00:07:03.220 Let's do one more problem. 00:07:03.220 --> 00:07:04.930 And this one I am just going to cut to the chase. 00:07:04.930 --> 00:07:15.250 Let's just solve x squared minus x minus 30 is equal to 0. 00:07:15.250 --> 00:07:19.550 Once again, two numbers when we add them they equal-- whats the 00:07:19.550 --> 00:07:21.850 coefficient here, it's negative 1. 00:07:21.850 --> 00:07:26.090 So we could say those two numbers are a plus b equals 00:07:26.090 --> 00:07:33.760 minus 1 and a times b will equal minus 30. 00:07:33.760 --> 00:07:37.050 Well let's just think about what all the factors are of 30. 00:07:37.050 --> 00:07:46.590 1, 2, 3, 5, 6, 10, 15, 30. 00:07:46.590 --> 00:07:50.930 Well, something interesting is happening this time though. 00:07:50.930 --> 00:07:55.320 Since a times b is negative 30, one of these numbers 00:07:55.320 --> 00:07:56.000 have to be negative. 00:07:56.000 --> 00:07:59.370 They both can't be negative, because if they're both 00:07:59.370 --> 00:08:01.800 negative then this would be a positive 30. 00:08:01.800 --> 00:08:04.950 a times b is negative 30. 00:08:04.950 --> 00:08:07.620 So actually we're going to have to say, two of these factors, 00:08:07.620 --> 00:08:12.030 the difference between them should be negative 1. 00:08:12.030 --> 00:08:13.930 Well, if we look at all of these, all these numbers 00:08:13.930 --> 00:08:17.850 obviously when you pair them up, they multiply out to 30. 00:08:17.850 --> 00:08:22.200 But the only ones that have a difference of 1 is 5 and 6. 00:08:22.200 --> 00:08:25.700 And since it's a negative 1, it's going to be-- and I know 00:08:25.700 --> 00:08:27.770 I'm going very fast with this and I'll do more example 00:08:27.770 --> 00:08:33.280 problems --this would be x minus 6 times x plus 00:08:33.280 --> 00:08:36.010 5 is equal to 0. 00:08:36.010 --> 00:08:37.850 So how did I think about that? 00:08:37.850 --> 00:08:42.160 Negative 6 times 5 is negative 30. 00:08:42.160 --> 00:08:46.180 Negative 6 plus 5 is negative 1. 00:08:46.180 --> 00:08:47.550 So it works out. 00:08:47.550 --> 00:08:50.000 And the more and more you do these practices-- I know it 00:08:50.000 --> 00:08:51.570 seems a little confusing right now --it'll make 00:08:51.570 --> 00:08:52.900 a lot more sense. 00:08:52.900 --> 00:09:00.010 So you get x equals 6 or x equals negative 5. 00:09:00.010 --> 00:09:04.150 I think at this point you're ready to try some solving 00:09:04.150 --> 00:09:06.610 quadratics by factoring and I'll do a couple more modules 00:09:06.610 --> 00:09:08.980 as soon as you get some more practice problems. 00:09:08.980 --> 00:09:10.520 Have fun.

Algebra: Linear equations 1

https://www.youtube.com/watch?v=bAerID24QJ0

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en

WEBVTT Kind: captions Language: en 00:00:01.230 --> 00:00:03.860 Welcome to level one linear equations. 00:00:03.860 --> 00:00:05.990 So let's start doing some problems. 00:00:05.990 --> 00:00:16.190 So let's say I had the equation 5-- a big fat 5, 5x equals 20. 00:00:16.190 --> 00:00:18.540 So at first this might look a little unfamiliar for you, 00:00:18.540 --> 00:00:19.914 but if I were to rephrase this, I 00:00:19.914 --> 00:00:22.920 think you'll realize this is a pretty easy problem. 00:00:22.920 --> 00:00:31.170 This is the same thing as saying 5 times question mark 00:00:31.170 --> 00:00:32.920 equals 20. 00:00:32.920 --> 00:00:34.920 And the reason we do the notation a little bit-- 00:00:34.920 --> 00:00:37.400 we write the 5 next to the x, because when 00:00:37.400 --> 00:00:39.300 you write a number right next to a variable, 00:00:39.300 --> 00:00:41.030 you assume that you're multiplying them. 00:00:41.030 --> 00:00:44.060 So this is just saying 5 times x, 00:00:44.060 --> 00:00:46.420 so instead of a question mark, we're writing an x. 00:00:46.420 --> 00:00:49.520 So 5 times x is equal to 20. 00:00:49.520 --> 00:00:51.520 Now, most of you all could do that in your head. 00:00:51.520 --> 00:00:53.990 You could say, well, what number times 5 is equal to 20? 00:00:53.990 --> 00:00:56.440 Well, it equals 4. 00:00:56.440 --> 00:00:58.960 But I'll show you a way to do it systematically just in case 00:00:58.960 --> 00:01:01.570 that 5 was a more complicated number. 00:01:01.570 --> 00:01:05.800 So let me make my pen a little thinner, OK. 00:01:05.800 --> 00:01:12.434 So rewriting it, if I had 5x equals 20, 00:01:12.434 --> 00:01:14.350 we could do two things and they're essentially 00:01:14.350 --> 00:01:15.100 the same thing. 00:01:15.100 --> 00:01:18.000 We could say we just divide both sides of this equation 00:01:18.000 --> 00:01:23.620 by 5, in which case, the left hand side, those two 5's will 00:01:23.620 --> 00:01:25.490 cancel out, we'll get x. 00:01:25.490 --> 00:01:28.590 And the right hand side, 20 divided by 5 is 4, 00:01:28.590 --> 00:01:29.870 and we would have solved it. 00:01:29.870 --> 00:01:32.260 Another way to do it, and this is actually the exact same way, 00:01:32.260 --> 00:01:34.010 we're just phrasing it a little different. 00:01:34.010 --> 00:01:40.950 If you said 5x equals 20, instead of dividing by 5, 00:01:40.950 --> 00:01:42.500 we could multiply by 1/5. 00:01:45.450 --> 00:01:48.620 And if you look at that, you can realize that multiplying by 1/5 00:01:48.620 --> 00:01:51.159 is the same thing as dividing by 5, 00:01:51.159 --> 00:01:52.950 if you know the difference between dividing 00:01:52.950 --> 00:01:54.560 and multiplying fractions. 00:01:54.560 --> 00:01:56.710 And then that gets the same thing, 1/5 times 5 00:01:56.710 --> 00:02:01.830 is 1, so you're just left with an x equals 4. 00:02:01.830 --> 00:02:03.880 I tend to focus a little bit more on this 00:02:03.880 --> 00:02:07.120 because when we start having fractions instead of a 5, 00:02:07.120 --> 00:02:09.980 it's easier just to think about multiplying by the reciprocal. 00:02:09.980 --> 00:02:11.730 Actually, let's do one of those right now. 00:02:14.260 --> 00:02:26.925 So let's say I had negative 3/4 times x equals 10/13. 00:02:30.060 --> 00:02:31.330 Now, this is a harder problem. 00:02:31.330 --> 00:02:32.820 I can't do this one in my head. 00:02:32.820 --> 00:02:36.890 We're saying negative 3/4 times some number 00:02:36.890 --> 00:02:39.412 x is equal to 10/13. 00:02:39.412 --> 00:02:41.870 If someone came up to you on the street and asked you that, 00:02:41.870 --> 00:02:46.040 I think you'd be like me, and you'd be pretty stumped. 00:02:46.040 --> 00:02:47.980 But let's work it out algebraically. 00:02:47.980 --> 00:02:49.710 Well, we do the same thing. 00:02:49.710 --> 00:02:53.930 We multiply both sides by the coefficient on x. 00:02:53.930 --> 00:02:57.390 So the coefficient, all that is, all that fancy word means, 00:02:57.390 --> 00:03:00.820 is the number that's being multiplied by x. 00:03:00.820 --> 00:03:04.390 So what's the reciprocal of minus 3/4. 00:03:04.390 --> 00:03:12.470 Well, it's minus 4/3 times, and dot is another way 00:03:12.470 --> 00:03:14.070 to use times, and you're probably 00:03:14.070 --> 00:03:17.624 wondering why in algebra, there are all these other conventions 00:03:17.624 --> 00:03:19.040 for doing times as opposed to just 00:03:19.040 --> 00:03:20.840 the traditional multiplication sign. 00:03:20.840 --> 00:03:23.800 And the main reason is, I think, just a regular multiplication 00:03:23.800 --> 00:03:25.860 sign gets confused with the variable x, 00:03:25.860 --> 00:03:28.770 so they thought of either using a dot if you're multiplying two 00:03:28.770 --> 00:03:31.170 constants, or just writing it next to a variable 00:03:31.170 --> 00:03:33.790 to imply you're multiplying a variable. 00:03:33.790 --> 00:03:36.460 So if we multiply the left hand side by negative 4/3, 00:03:36.460 --> 00:03:38.090 we also have to do the same thing 00:03:38.090 --> 00:03:42.280 to the right hand side, minus 4/3. 00:03:42.280 --> 00:03:44.850 The left hand side, the minus 4/3 and the 3/4, 00:03:44.850 --> 00:03:46.280 they cancel out. 00:03:46.280 --> 00:03:48.920 You could work it out on your own to see that they do. 00:03:48.920 --> 00:03:53.610 They equal 1, so we're just left with x is equal to 10 times 00:03:53.610 --> 00:04:01.740 minus 4 is minus 40, 13 times 3, well, that's equal to 39. 00:04:01.740 --> 00:04:04.180 So we get x is equal to minus 40/39. 00:04:07.100 --> 00:04:08.930 And I like to leave my fractions improper 00:04:08.930 --> 00:04:11.170 because it's easier to deal with them. 00:04:11.170 --> 00:04:14.185 But you could also view that-- that's minus-- 00:04:14.185 --> 00:04:16.310 if you wanted to write it as a mixed number, that's 00:04:16.310 --> 00:04:17.130 minus 1 and 1/39. 00:04:19.839 --> 00:04:22.220 I tend to keep it like this. 00:04:22.220 --> 00:04:24.160 Let's check to make sure that's right. 00:04:24.160 --> 00:04:26.770 The cool thing about algebra is you can always get your answer 00:04:26.770 --> 00:04:28.230 and put it back into the original equation 00:04:28.230 --> 00:04:29.355 to make sure you are right. 00:04:29.355 --> 00:04:34.860 So the original equation was minus 3/4 times x, 00:04:34.860 --> 00:04:38.940 and here we'll substitute the x back into the equation. 00:04:38.940 --> 00:04:43.150 Wherever we saw x, we'll now put our answer. 00:04:43.150 --> 00:04:49.550 So it's minus 40/39, and our original equation 00:04:49.550 --> 00:04:52.786 said that equals 10/13. 00:04:52.786 --> 00:04:54.170 Well, and once again, when I just 00:04:54.170 --> 00:04:57.260 write the 3/4 right next to the parentheses like that, 00:04:57.260 --> 00:05:00.060 that's just another way of writing times. 00:05:00.060 --> 00:05:08.132 So minus 3 times minus 40, it is minus 100-- 00:05:08.132 --> 00:05:10.340 Actually, we could do something a little bit simpler. 00:05:10.340 --> 00:05:16.370 This 4 becomes a 1 and this becomes a 10. 00:05:16.370 --> 00:05:18.740 If you remember when you're multiplying fractions, 00:05:18.740 --> 00:05:21.480 you can simplify it like that. 00:05:21.480 --> 00:05:26.390 So it actually becomes minus-- actually, plus 30, 00:05:26.390 --> 00:05:30.970 because we have a minus times a minus and 3 times 10, over, 00:05:30.970 --> 00:05:34.550 the 4 is now 1, so all we have left is 39. 00:05:34.550 --> 00:05:39.250 And 30/39, if we divide the top and the bottom by 3, 00:05:39.250 --> 00:05:45.130 we get 10 over 13, which is the same thing as what 00:05:45.130 --> 00:05:47.050 the equation said we would get, so we 00:05:47.050 --> 00:05:49.597 know that we've got the right answer. 00:05:49.597 --> 00:05:50.680 Let's do one more problem. 00:05:54.750 --> 00:06:01.532 Minus 5/6x is equal to 7/8. 00:06:01.532 --> 00:06:02.990 And if you want to try this problem 00:06:02.990 --> 00:06:05.570 yourself, now's a good time to pause, 00:06:05.570 --> 00:06:08.400 and I'm going to start doing the problem right now. 00:06:08.400 --> 00:06:10.010 So same thing. 00:06:10.010 --> 00:06:12.090 What's the reciprocal of minus 5/6? 00:06:12.090 --> 00:06:15.700 Well, it's minus 6/5. 00:06:15.700 --> 00:06:16.710 We multiply that. 00:06:16.710 --> 00:06:18.620 If you do that on the left hand side, 00:06:18.620 --> 00:06:21.750 we have to do it on the right hand side as well. 00:06:21.750 --> 00:06:23.430 Minus 6/5. 00:06:23.430 --> 00:06:26.920 The left hand side, the minus 6/5 and the minus 5/6 00:06:26.920 --> 00:06:27.660 cancel out. 00:06:27.660 --> 00:06:30.030 We're just left with x. 00:06:30.030 --> 00:06:33.150 And the right hand side, we have, 00:06:33.150 --> 00:06:37.740 well, we can divide both the 6 and the 8 by 2, 00:06:37.740 --> 00:06:40.330 so this 6 becomes negative 3. 00:06:40.330 --> 00:06:42.882 This becomes 4. 00:06:42.882 --> 00:06:45.575 7 times negative 3 is minus 21/20. 00:06:50.087 --> 00:06:52.170 And assuming I haven't made any careless mistakes, 00:06:52.170 --> 00:06:53.045 that should be right. 00:06:53.045 --> 00:06:55.340 Actually, let's just check that real quick. 00:06:55.340 --> 00:06:59.580 Minus 5/6 times minus 21/20. 00:07:02.350 --> 00:07:06.672 Well, that equals 5, make that into 1. 00:07:06.672 --> 00:07:08.940 Turn this into a 4. 00:07:08.940 --> 00:07:10.205 Make this into a 2. 00:07:10.205 --> 00:07:12.680 Make this into a 7. 00:07:12.680 --> 00:07:14.280 Negative times negative is positive. 00:07:14.280 --> 00:07:15.710 So you have 7. 00:07:15.710 --> 00:07:17.640 2 times 4 is 8. 00:07:17.640 --> 00:07:19.615 And that's what we said we would get. 00:07:19.615 --> 00:07:21.290 So we got it right. 00:07:21.290 --> 00:07:22.710 I think you're ready at this point 00:07:22.710 --> 00:07:25.220 to try some level one equations. 00:07:25.220 --> 00:07:27.040 Have fun.

Algebra: Linear equations 2

https://www.youtube.com/watch?v=DopnmxeMt-s

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https://www.youtube.com/api/timedtext?v=DopnmxeMt-s&ei=g2eUZeLHOJ21vdIPk72AwAw&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249843&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=7AEBE399BC8700241C8473003CAAE7437DEA5C2B.A3065F78C29C6EC25CE1A629A43BE153E0633D8F&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.340 --> 00:00:03.440 Welcome to level two linear equations. 00:00:03.440 --> 00:00:05.330 Let's do a problem. 00:00:05.330 --> 00:00:14.130 2x plus 3 is equal to minus 15. 00:00:14.130 --> 00:00:17.260 Throw the minus in there to make it a little bit tougher. 00:00:17.260 --> 00:00:18.970 So the first thing we want to do whenever 00:00:18.970 --> 00:00:20.850 we do any linear equation, is we want 00:00:20.850 --> 00:00:23.190 to get all of the variable terms on one hand 00:00:23.190 --> 00:00:25.180 side of the equation and all the constant terms 00:00:25.180 --> 00:00:25.930 on the other side. 00:00:25.930 --> 00:00:27.860 And it doesn't really matter, although I 00:00:27.860 --> 00:00:30.970 tend to get my variables on the left hand side of the equation. 00:00:30.970 --> 00:00:34.530 Well, my variables are already on the left hand 00:00:34.530 --> 00:00:36.570 side of the equation but I have this plus 3 00:00:36.570 --> 00:00:39.020 that I somehow want to move to the right hand 00:00:39.020 --> 00:00:40.470 side of the equation. 00:00:40.470 --> 00:00:43.900 And the way I can-- you can put it in quotes, move the 3 is I 00:00:43.900 --> 00:00:48.080 can subtract 3 from both sides of this equation. 00:00:48.080 --> 00:00:51.690 And look at that carefully as to why you think that works. 00:00:51.690 --> 00:00:53.690 Because if I subtract 3 from the left hand side, 00:00:53.690 --> 00:00:57.350 clearly this negative 3 that I'm subtracting and the original 3 00:00:57.350 --> 00:00:59.690 will cancel out and become 0. 00:00:59.690 --> 00:01:02.250 and as long as I do whatever I do on the left hand side, 00:01:02.250 --> 00:01:03.958 as long as I do it on the right hand side 00:01:03.958 --> 00:01:07.526 as well, because whatever you do on one side of the equal side, 00:01:07.526 --> 00:01:08.900 you have to do to the other side, 00:01:08.900 --> 00:01:11.130 then I'm making a valid operation. 00:01:11.130 --> 00:01:15.600 So this will simplify to 2x, because the 3's cancel out. 00:01:15.600 --> 00:01:17.200 They become just 0. 00:01:17.200 --> 00:01:20.320 Equals minus 15 minus 3. 00:01:20.320 --> 00:01:22.595 Well, that's minus 18. 00:01:25.200 --> 00:01:28.200 And now, we're just at a level one problem, 00:01:28.200 --> 00:01:29.840 and you can just multiply both sides 00:01:29.840 --> 00:01:31.630 of this equation times the reciprocal 00:01:31.630 --> 00:01:32.890 on the coefficient of 2x. 00:01:35.850 --> 00:01:37.274 I mean, some people would just say 00:01:37.274 --> 00:01:39.190 that we're dividing by 2, which is essentially 00:01:39.190 --> 00:01:40.140 what we're doing. 00:01:40.140 --> 00:01:42.020 I like to always go with the reciprocal, 00:01:42.020 --> 00:01:44.410 because if this 2 was a fraction, 00:01:44.410 --> 00:01:46.270 it's easier to think about it that way. 00:01:46.270 --> 00:01:49.250 But either way, you either multiply by the reciprocal, 00:01:49.250 --> 00:01:50.250 or divide by the number. 00:01:50.250 --> 00:01:51.580 It's the same thing. 00:01:51.580 --> 00:01:52.590 So 1/2 times 2x. 00:01:52.590 --> 00:01:54.930 Well, that's just 1x. 00:01:54.930 --> 00:02:00.780 So you get x equals, and then minus 18/2. 00:02:00.780 --> 00:02:05.660 And minus 18/2, well, that just equals minus 9. 00:02:05.660 --> 00:02:07.060 Let's do another problem. 00:02:07.060 --> 00:02:10.530 And actually, well, if we wanted to check it, we could say, 00:02:10.530 --> 00:02:17.070 well, the original problem was 2x plus 3 equals minus 15. 00:02:17.070 --> 00:02:22.160 So we could say 2 times minus 9 plus 3. 00:02:22.160 --> 00:02:25.530 2 times minus 9 is minus 18 plus 3. 00:02:25.530 --> 00:02:29.090 Well, that's equal to minus 15, which 00:02:29.090 --> 00:02:31.120 is equal to what the original equation said, 00:02:31.120 --> 00:02:32.120 so we know that's right. 00:02:32.120 --> 00:02:33.620 That's the neat thing about algebra. 00:02:33.620 --> 00:02:35.279 You can always check your work. 00:02:35.279 --> 00:02:36.320 Let's do another problem. 00:02:36.320 --> 00:02:38.560 I'm going to put some fractions in this time, 00:02:38.560 --> 00:02:43.360 just to show you that it can get a little bit hairy. 00:02:43.360 --> 00:02:57.135 So let's say I had minus 1/2x plus 3/4 is equal to 5/6. 00:02:57.135 --> 00:02:58.260 So we'll do the same thing. 00:02:58.260 --> 00:03:01.460 First, we just want to get this 3/4 out of the left hand 00:03:01.460 --> 00:03:03.846 side of the equation, and actually, 00:03:03.846 --> 00:03:05.720 if you want to try working this out yourself, 00:03:05.720 --> 00:03:06.650 you might want to pause the video 00:03:06.650 --> 00:03:09.240 and then play it once you're ready to see how I do it. 00:03:09.240 --> 00:03:12.360 Anyway, let me move forward assuming you haven't paused it. 00:03:12.360 --> 00:03:14.730 If we want to get rid of this 3/4, all we do 00:03:14.730 --> 00:03:18.175 is we subtract 3/4 from both sides of this equation. 00:03:20.740 --> 00:03:23.240 Minus 3/4. 00:03:23.240 --> 00:03:26.990 Well, the left hand side, the two 3/4 will just cancel. 00:03:26.990 --> 00:03:33.640 We get minus 1/2x equals, and then on the right hand side, 00:03:33.640 --> 00:03:38.200 we just have to do this fraction addition or fraction 00:03:38.200 --> 00:03:38.880 subtraction. 00:03:38.880 --> 00:03:44.120 So the least common multiple of 6 and 4 is 12. 00:03:44.120 --> 00:03:53.470 So this becomes 5/6 6 is 10/12 minus 3/4 is 9/12, 00:03:53.470 --> 00:04:01.900 so we get minus 1/2x is equal to 1/12. 00:04:01.900 --> 00:04:04.190 Hopefully, I didn't make a mistake over here. 00:04:04.190 --> 00:04:06.490 And if that step confused you, I went a little fast, 00:04:06.490 --> 00:04:08.156 you might just want to review the adding 00:04:08.156 --> 00:04:10.009 and subtraction of fractions. 00:04:10.009 --> 00:04:11.300 So going back to where we were. 00:04:11.300 --> 00:04:13.400 So now all we have to do is, well, the coefficient 00:04:13.400 --> 00:04:16.850 on the x term is minus 1/2, and this is now a level one 00:04:16.850 --> 00:04:17.490 problem. 00:04:17.490 --> 00:04:19.490 So to solve for x, we just multiply 00:04:19.490 --> 00:04:23.160 both sides by the reciprocal of this minus 1/2x, 00:04:23.160 --> 00:04:29.570 and that's minus 2/1 times minus 1/2x on that side, 00:04:29.570 --> 00:04:34.880 and then that's times minus 2/1. 00:04:34.880 --> 00:04:37.430 The left hand side, and you're used to this by now, 00:04:37.430 --> 00:04:40.180 simplifies to x. 00:04:40.180 --> 00:04:45.510 The right hand side becomes minus 2/12, 00:04:45.510 --> 00:04:49.160 and we could simplify that further to minus 1/6. 00:04:49.160 --> 00:04:55.750 Well, let's check that just to make sure we got it right. 00:04:55.750 --> 00:04:58.840 So let's try to remember that minus 1/6. 00:04:58.840 --> 00:05:03.640 So the original problem was minus 1/2x, 00:05:03.640 --> 00:05:08.270 so here we can substitute the minus 1/6, plus 3/4. 00:05:08.270 --> 00:05:12.150 I just wrote only the left hand side of the original problem. 00:05:12.150 --> 00:05:15.640 So minus 1/2 times minus 1/6, well, that's 00:05:15.640 --> 00:05:23.150 positive 1/12 plus 3/4. 00:05:23.150 --> 00:05:25.730 Well, that's the same thing as 12, 00:05:25.730 --> 00:05:29.270 the 1 stays the same, plus 9. 00:05:29.270 --> 00:05:33.530 1 plus 9 is 10 over 12. 00:05:33.530 --> 00:05:36.650 And that is equal to 5/6, which is 00:05:36.650 --> 00:05:38.020 what our original problem was. 00:05:38.020 --> 00:05:38.730 Our original problem was this. 00:05:38.730 --> 00:05:39.850 This stuff I wrote later. 00:05:39.850 --> 00:05:43.150 So it's 5/6, so the problem checks out. 00:05:43.150 --> 00:05:46.750 So hopefully, you're now ready to try some level two 00:05:46.750 --> 00:05:48.664 problems on your own. 00:05:48.664 --> 00:05:50.330 I might add some other example problems. 00:05:50.330 --> 00:05:55.000 But the only extra step here relative to level one problems 00:05:55.000 --> 00:05:58.062 is you'll have this constant term that you 00:05:58.062 --> 00:06:00.930 need to add or subtract from both sides of this equation, 00:06:00.930 --> 00:06:03.720 and you'll essentially turn it into a level one problem. 00:06:03.720 --> 00:06:05.329 Have fun.

Algebra: Linear equations 3

https://www.youtube.com/watch?v=Zn-GbH2S0Dk

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en

WEBVTT Kind: captions Language: en 00:00:01.730 --> 00:00:03.620 Welcome to my presentation on level 00:00:03.620 --> 00:00:07.115 three linear, yeah, level three linear equations. 00:00:07.115 --> 00:00:08.280 [LAUGH] Okay. 00:00:08.280 --> 00:00:10.510 So let me, let's, let's make up a problem. 00:00:10.510 --> 00:00:17.709 Let's say I had X plus 2x plus 3 is 00:00:17.709 --> 00:00:25.170 equal to, minus 7x minus 5. 00:00:25.170 --> 00:00:29.530 Well, in all of these linear equations, the first things that 00:00:29.530 --> 00:00:31.600 we, the first thing that we try to do is, get all 00:00:31.600 --> 00:00:34.480 of our variables on one side of the equation, and then get 00:00:34.480 --> 00:00:36.620 all of our concept terms on the other side of the equation. 00:00:36.620 --> 00:00:39.950 And then it actually will become a level one linear equation. 00:00:39.950 --> 00:00:44.050 So, the first thing we can do is we can try to simplify each of the sides. 00:00:44.050 --> 00:00:47.240 Well we, on this, on this left side we have this X plus 2x. 00:00:47.240 --> 00:00:48.610 Well, what is X plus 2x? 00:00:48.610 --> 00:00:51.870 Well that's like saying I have one apple and now I have two apples. 00:00:51.870 --> 00:00:55.090 So here I have one X and now I have two more Xs that I'm adding together. 00:00:55.090 --> 00:01:03.650 So that's equal 3x, 3x plus 3 is equal to minus 7x minus 5. 00:01:03.650 --> 00:01:07.310 Now let's bring the 7x over onto the left-hand side. 00:01:07.310 --> 00:01:11.840 And we could do that by adding 7x to both sides, 7x. 00:01:11.840 --> 00:01:13.733 This is a review. 00:01:13.733 --> 00:01:16.882 We, we're adding the opposite. 00:01:16.882 --> 00:01:21.290 So, it's negative 7x, so we add 7x so that's why. 00:01:21.290 --> 00:01:25.620 And we do that, become the right side, these two will cancel. 00:01:27.990 --> 00:01:33.390 And the left side, we get 10x plus 00:01:33.390 --> 00:01:40.390 3 equals, and on the right side, all we have left is the negative 5. 00:01:40.390 --> 00:01:43.040 Almost there, now we're at a level, what is this, a level two problem. 00:01:43.040 --> 00:01:44.940 And now we just have to take this 3 and move it to the other side. 00:01:44.940 --> 00:01:47.590 And we can do that by subtracting 3 from both sides. 00:01:52.650 --> 00:01:53.150 That's a 3 minus 3. 00:01:53.150 --> 00:01:57.390 The left-hand side, the 3s cancel out, that's 00:01:57.390 --> 00:02:00.550 why we subtract it in the first place. 00:02:00.550 --> 00:02:06.010 And you have 10x equals and then minus 5 minus 3, well that equals minus 8. 00:02:06.010 --> 00:02:09.970 Now, we just multiply both sides of this equation by 1 over 10, or 00:02:09.970 --> 00:02:14.480 the reciprocal of 10, which is the coefficient on x, times 1 over 10. 00:02:14.480 --> 00:02:16.860 You could also, some people would say, well, we're just 00:02:16.860 --> 00:02:19.020 dividing both side by 10 which is essential what we're doing. 00:02:19.020 --> 00:02:22.226 If you divide by 10, that's the same thing as multiplying by 1 over 10. 00:02:22.226 --> 00:02:25.580 Well, anyway, the left-hand side, 1 over 10 times 10. 00:02:25.580 --> 00:02:30.930 Well, that equals 1, so we're just left with x equals negative 8 over 10. 00:02:30.930 --> 00:02:33.830 And that can be reduced further. 00:02:33.830 --> 00:02:35.440 They both share the common factor 2. 00:02:35.440 --> 00:02:37.360 So you divide by 2. 00:02:37.360 --> 00:02:38.760 So it's minus 4 over 5. 00:02:38.760 --> 00:02:45.620 I think that's right, assuming that I didn't make any careless mistakes. 00:02:45.620 --> 00:02:49.288 Let's do another problem. 00:02:49.288 --> 00:02:54.321 Let's say I had 5, 00:02:54.321 --> 00:03:01.133 that's a 5x minus 3 minus 00:03:01.133 --> 00:03:06.474 7x equals x plus 8. 00:03:07.570 --> 00:03:10.778 And in general if you wanna work this out before I give you 00:03:10.778 --> 00:03:14.340 how I do it that now's a good time to actually pause the video. 00:03:14.340 --> 00:03:17.028 And you could, you could try to work it out and then, 00:03:17.028 --> 00:03:19.548 play it again and, and see what I have to say about it. 00:03:19.548 --> 00:03:22.284 But assuming you wanna hear it, let me go and do it. 00:03:22.284 --> 00:03:23.450 So let's do the same thing. 00:03:23.450 --> 00:03:26.390 We, first of all, we can merge these two Xs on the left-hand side. 00:03:26.390 --> 00:03:28.710 Remember, you can't add the 5 and the 3 because the 3 00:03:28.710 --> 00:03:31.160 is just a constant term while the 5 is 5 times x. 00:03:32.180 --> 00:03:35.720 But the 5 times x and the negative 7 times actually can merge. 00:03:35.720 --> 00:03:38.570 So 5, you just add the coefficient. 00:03:38.570 --> 00:03:41.160 So, it's 5 and negative 7. 00:03:41.160 --> 00:03:51.110 So, that becomes negative 2x minus 3 is equal to x plus 8. 00:03:51.110 --> 00:03:53.760 Now, if we wanna take this x that's on the right-hand side 00:03:53.760 --> 00:03:56.810 and put it over the left-hand side, we can just subtract x from 00:04:00.690 --> 00:04:01.002 both sides. 00:04:01.002 --> 00:04:04.030 The left-hand side becomes minus 3x minus 3 is equal 00:04:04.030 --> 00:04:10.550 to, these two Xs cancel out, is equal to 8. 00:04:10.550 --> 00:04:12.050 Now, we can just add 3 to both sides to 00:04:12.050 --> 00:04:16.810 get rid of that constant term 3 on left hand-side. 00:04:16.810 --> 00:04:18.649 These two 3's will cancel out. 00:04:18.649 --> 00:04:22.300 And you get minus 3x is equal to 11. 00:04:22.300 --> 00:04:29.510 Now, you just multiply both sides by negative one-third. 00:04:29.510 --> 00:04:30.670 And once again, this is just the same 00:04:30.670 --> 00:04:34.750 thing as dividing both sides by negative 3. 00:04:34.750 --> 00:04:38.320 And you get x equals negative 11 over 3. 00:04:38.320 --> 00:04:38.870 Actually 00:04:41.620 --> 00:04:44.670 let's, let's, just for fun, let's check this just to see. 00:04:44.670 --> 00:04:47.354 And the cool thing about algebra is if you have enough 00:04:47.354 --> 00:04:50.296 time, you can always make sure you got the right answer. 00:04:50.296 --> 00:04:52.450 So we have 5x, so we have 5 times negative 11 over 3. 00:04:52.450 --> 00:04:55.380 So that's, I'm just, I'm just gonna take 00:04:57.230 --> 00:05:02.660 this and substitute it back into the original equation. 00:05:02.660 --> 00:05:06.410 And you might wanna try that out, too. 00:05:06.410 --> 00:05:09.190 So you have minus 55 over 3, that's just 5 00:05:09.190 --> 00:05:14.860 times negative 11 over 3, that's a 3, minus 3. 00:05:14.860 --> 00:05:15.625 And what's 3? 00:05:15.625 --> 00:05:17.920 Three could also be written as, minus 9 over 3. 00:05:17.920 --> 00:05:23.428 I'm skipping some steps, but I think 00:05:23.428 --> 00:05:28.440 you, you know your fractions pretty good by this point. 00:05:28.440 --> 00:05:29.730 So that's minus 9 over 3. 00:05:29.730 --> 00:05:32.420 And then, minus 7x is the same thing, as plus 77 over 3. 00:05:32.420 --> 00:05:38.090 Because we have the minus 7 times minus 11, so it's plus 77. 00:05:38.090 --> 00:05:45.240 And, and the equation is saying that should equal minus 11 over 00:05:45.240 --> 00:05:52.650 3, that's what x is, plus an 8 is nothing more than 24 over 3. 00:05:52.650 --> 00:05:53.340 Let's add this up. 00:05:53.340 --> 00:06:01.570 Minus 55 minus 9, that's minus 64, if I'm right, yeah, that's minus 64. 00:06:01.570 --> 00:06:08.780 And then, plus 77 minus 64 plus 77 is 13. 00:06:08.780 --> 00:06:13.084 So the left-hand side becomes 13 over 3. 00:06:14.160 --> 00:06:15.540 And on the right-hand side minus 11 plus 24, 00:06:15.540 --> 00:06:16.840 well that's 13 and we still have over 3. 00:06:16.840 --> 00:06:20.836 So looks like we got the right solution. 00:06:20.836 --> 00:06:23.610 It checks out. 00:06:23.610 --> 00:06:28.180 So the correct answer was minus 11 over 3. 00:06:28.180 --> 00:06:32.220 Hopefully you're ready by now to, do some level three problems. 00:06:32.220 --> 00:06:34.230 The only thing that makes this a little bit 00:06:34.230 --> 00:06:36.620 more complicated than level two is you just have to 00:06:36.620 --> 00:06:39.030 remember to merge the variables in the beginning, and, know 00:06:39.030 --> 00:06:41.840 that you could subtract variables or constants from both sides. 00:06:41.840 --> 00:06:42.070 Have fun.

Algebra: Linear equations 4

https://www.youtube.com/watch?v=9IUEk9fn2Vs

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en

WEBVTT Kind: captions Language: en 00:00:01.230 --> 00:00:04.280 Welcome to the presentation on level four linear equations. 00:00:04.280 --> 00:00:06.540 So, let's start doing some problems. 00:00:06.540 --> 00:00:06.710 So. 00:00:06.710 --> 00:00:09.580 Let's say I had the situation-- let me give me a couple of 00:00:09.580 --> 00:00:20.110 problems-- if I said 3 over x is equal to, let's just say 5. 00:00:20.110 --> 00:00:23.180 So, what we want to do -- this problem's a little unusual from 00:00:23.180 --> 00:00:24.260 everything we've ever seen. 00:00:24.260 --> 00:00:26.950 Because here, instead of having x in the numerator, we actually 00:00:26.950 --> 00:00:28.150 have x in the denominator. 00:00:28.150 --> 00:00:31.270 So, I personally don't like having x's in my denominators, 00:00:31.270 --> 00:00:34.190 so we want to get it outside of the denominator into a 00:00:34.190 --> 00:00:36.140 numerator or at least not in the denominator as 00:00:36.140 --> 00:00:36.920 soon as possible. 00:00:36.920 --> 00:00:40.780 So, one way to get a number out of the denominator is, if we 00:00:40.780 --> 00:00:45.560 were to multiply both sides of this equation by x, you see 00:00:45.560 --> 00:00:47.460 that on the left-hand side of the equation these two 00:00:47.460 --> 00:00:48.900 x's will cancel out. 00:00:48.900 --> 00:00:52.160 And in the right side, you'll just get 5 times x. 00:00:52.160 --> 00:00:56.920 So this equals -- the two x's cancel out. 00:00:56.920 --> 00:01:00.890 And you get 3 is equal to 5x. 00:01:00.890 --> 00:01:05.420 Now, we could also write that as 5x is equal to 3. 00:01:05.420 --> 00:01:07.810 And then we can think about this two ways. 00:01:07.810 --> 00:01:12.210 We either just multiply both sides by 1/5, or you could just 00:01:12.210 --> 00:01:14.230 do that as dividing by 5. 00:01:14.230 --> 00:01:16.490 If you multiply both sides by 1/5. 00:01:16.490 --> 00:01:18.680 The left-hand side becomes x. 00:01:18.680 --> 00:01:23.740 And the right-hand side, 3 times 1/5, is equal to 3/5. 00:01:23.740 --> 00:01:24.640 So what did we do here? 00:01:24.640 --> 00:01:26.860 This is just like, this actually turned into a level 00:01:26.860 --> 00:01:28.670 two problem, or actually a level one problem, 00:01:28.670 --> 00:01:29.480 very quickly. 00:01:29.480 --> 00:01:31.990 All we had to do is multiply both sides of this 00:01:31.990 --> 00:01:33.260 equation by x. 00:01:33.260 --> 00:01:35.460 And we got the x's out of the denominator. 00:01:35.460 --> 00:01:36.360 Let's do another problem. 00:01:41.110 --> 00:01:53.530 Let's have -- let me say, x plus 2 over x plus 1 is 00:01:53.530 --> 00:01:58.800 equal to, let's say, 7. 00:01:58.800 --> 00:02:00.790 So, here, instead of having just an x in the denominator, 00:02:00.790 --> 00:02:02.920 we have a whole x plus 1 in the denominator. 00:02:02.920 --> 00:02:05.000 But we're going to do it the same way. 00:02:05.000 --> 00:02:09.170 To get that x plus 1 out of the denominator, we multiply both 00:02:09.170 --> 00:02:15.450 sides of this equation times x plus 1 over 1 times this side. 00:02:15.450 --> 00:02:17.010 Since we did it on the left-hand side we also have 00:02:17.010 --> 00:02:19.640 to do it on the right-hand side, and this is just 7/1, 00:02:19.640 --> 00:02:24.420 times x plus 1 over 1. 00:02:24.420 --> 00:02:27.720 On the left-hand side, the x plus 1's cancel out. 00:02:27.720 --> 00:02:31.110 And you're just left with x plus 2. 00:02:31.110 --> 00:02:33.300 It's over 1, but we can just ignore the 1. 00:02:33.300 --> 00:02:39.260 And that equals 7 times x plus 1. 00:02:39.260 --> 00:02:41.930 And that's the same thing as x plus 2. 00:02:41.930 --> 00:02:45.720 And, remember, it's 7 times the whole thing, x plus 1. 00:02:45.720 --> 00:02:47.790 So we actually have to use the distributive property. 00:02:47.790 --> 00:02:54.400 And that equals 7x plus 7. 00:02:54.400 --> 00:02:57.200 So now it's turned into a, I think this is a level 00:02:57.200 --> 00:02:58.790 three linear equation. 00:02:58.790 --> 00:03:02.050 And now all we do is, we say well let's get all the x's on 00:03:02.050 --> 00:03:02.965 one side of the equation. 00:03:02.965 --> 00:03:05.570 And let's get all the constant terms, like the 2 and the 7, on 00:03:05.570 --> 00:03:07.100 the other side of the equation. 00:03:07.100 --> 00:03:08.890 So I'm going to choose to get the x's on the left. 00:03:08.890 --> 00:03:10.990 So let's bring that 7x onto the left. 00:03:10.990 --> 00:03:14.450 And we can do that by subtracting 7x from both sides. 00:03:14.450 --> 00:03:19.440 Minus 7x, plus, it's a minus 7x. 00:03:19.440 --> 00:03:22.800 The right-hand side, these two 7x's will cancel out. 00:03:22.800 --> 00:03:26.410 And on the left-hand side we have minus 7x plus x. 00:03:26.410 --> 00:03:32.840 Well, that's minus 6x plus 2 is equal to, and on the 00:03:32.840 --> 00:03:35.080 right all we have left is 7. 00:03:35.080 --> 00:03:36.470 Now we just have to get rid of this 2. 00:03:36.470 --> 00:03:41.360 And we can just do that by subtracting 2 from both sides. 00:03:41.360 --> 00:03:48.000 And we're left with minus 6x packs is equal to 6. 00:03:48.000 --> 00:03:49.220 Now it's a level one problem. 00:03:49.220 --> 00:03:52.410 We just have to multiply both sides times the reciprocal 00:03:52.410 --> 00:03:54.200 of the coefficient on the left-hand side. 00:03:54.200 --> 00:03:56.150 And the coefficient's negative 6. 00:03:56.150 --> 00:03:59.620 So we multiply both sides of the equation by negative 1/6. 00:04:02.540 --> 00:04:05.610 Negative 1/6. 00:04:05.610 --> 00:04:08.890 The left-hand side, negative 1 over 6 times negative 6. 00:04:08.890 --> 00:04:10.190 Well that just equals 1. 00:04:10.190 --> 00:04:16.130 So we just get x is equal to 5 times negative 1/6. 00:04:16.130 --> 00:04:19.250 Well, that's negative 5/6. 00:04:22.270 --> 00:04:23.210 And we're done. 00:04:23.210 --> 00:04:25.710 And if you wanted to check it, you could just take that x 00:04:25.710 --> 00:04:28.950 equals negative 5/6 and put it back in the original question 00:04:28.950 --> 00:04:30.580 to confirm that it worked. 00:04:30.580 --> 00:04:31.340 Let's do another one. 00:04:34.610 --> 00:04:37.940 I'm making these up on the fly, so I apologize. 00:04:37.940 --> 00:04:40.020 Let me think. 00:04:40.020 --> 00:04:51.010 3 times x plus 5 is equal to 8 times x plus 2. 00:04:51.010 --> 00:04:52.740 Well, we do the same thing here. 00:04:52.740 --> 00:04:55.950 Although now we have two expressions we want to get 00:04:55.950 --> 00:04:56.680 out of the denominators. 00:04:56.680 --> 00:04:58.870 We want to get x plus 5 out and we want to get 00:04:58.870 --> 00:05:00.010 this x plus 2 out. 00:05:00.010 --> 00:05:01.670 So let's do the x plus 5 first. 00:05:01.670 --> 00:05:03.640 Well, just like we did before, we multiply both sides of 00:05:03.640 --> 00:05:05.570 this equation by x plus 5. 00:05:05.570 --> 00:05:07.630 You can say x plus 5 over 1. 00:05:07.630 --> 00:05:12.680 Times x plus 5 over 1. 00:05:12.680 --> 00:05:15.080 On the left-hand side, they get canceled out. 00:05:15.080 --> 00:05:24.230 So we're left with 3 is equal to 8 times x plus five. 00:05:24.230 --> 00:05:28.770 All of that over x plus 2. 00:05:28.770 --> 00:05:31.820 Now, on the top, just to simplify, we once again 00:05:31.820 --> 00:05:34.420 just multiply the 8 times the whole expression. 00:05:34.420 --> 00:05:41.860 So it's 8x plus 40 over x plus 2. 00:05:41.860 --> 00:05:43.500 Now, we want to get rid of this x plus 2. 00:05:43.500 --> 00:05:44.510 So we can do it the same way. 00:05:44.510 --> 00:05:46.505 We can multiply both sides of this equation by 00:05:46.505 --> 00:05:50.904 x plus 2 over 1. 00:05:50.904 --> 00:05:52.580 x plus 2. 00:05:52.580 --> 00:05:53.690 We could just say we're multiplying both 00:05:53.690 --> 00:05:54.420 sides by x plus 2. 00:05:54.420 --> 00:05:56.630 The 1 is little unnecessary. 00:05:56.630 --> 00:06:02.910 So the left-hand side becomes 3x plus 6. 00:06:02.910 --> 00:06:05.070 Remember, always distribute 3 times, because you're 00:06:05.070 --> 00:06:07.030 multiplying it times the whole expression. 00:06:07.030 --> 00:06:08.540 x plus 2. 00:06:08.540 --> 00:06:09.860 And on the right-hand side. 00:06:09.860 --> 00:06:13.620 Well, this x plus 2 and this x plus 2 will cancel out. 00:06:13.620 --> 00:06:16.380 And we're left with 8x plus 40. 00:06:16.380 --> 00:06:19.340 And this is now a level three problem. 00:06:19.340 --> 00:06:25.380 Well, if we subtract 8x from both sides, minus 8x, plus-- I 00:06:25.380 --> 00:06:26.970 think I'm running out of space. 00:06:26.970 --> 00:06:28.470 Minus 8x. 00:06:28.470 --> 00:06:31.290 Well, on the right-hand side the 8x's cancel out. 00:06:31.290 --> 00:06:38.620 On the left-hand side we have minus 5x plus 6 is equal 00:06:38.620 --> 00:06:42.320 to, on the right-hand side all we have left is 40. 00:06:42.320 --> 00:06:45.380 Now we can subtract 6 from both sides of this equation. 00:06:45.380 --> 00:06:46.380 Let me just write out here. 00:06:46.380 --> 00:06:49.510 Minus 6 plus minus 6. 00:06:49.510 --> 00:06:51.470 Now I'm going to, hope I don't lose you guys by 00:06:51.470 --> 00:06:53.160 trying to go up here. 00:06:55.720 --> 00:06:58.410 But if we subtract minus 6 from both sides, on the left-hand 00:06:58.410 --> 00:07:05.280 side we're just left with minus 5x equals, and on the 00:07:05.280 --> 00:07:08.780 right-hand side we have 34. 00:07:08.780 --> 00:07:09.880 Now it's a level one problem. 00:07:09.880 --> 00:07:12.780 We just multiply both sides times negative 1/5. 00:07:16.510 --> 00:07:18.360 Negative 1/5. 00:07:18.360 --> 00:07:21.130 On the left-hand side we have x. 00:07:21.130 --> 00:07:27.130 And on the right-hand side we have negative 34/5. 00:07:27.130 --> 00:07:29.640 Unless I made some careless mistakes, I think that's right. 00:07:29.640 --> 00:07:33.190 And I think if you understood what we just did here, you're 00:07:33.190 --> 00:07:36.780 ready to tackle some level four linear equations. 00:07:36.780 --> 00:07:38.290 Have fun.

Equivalent fractions

https://www.youtube.com/watch?v=U2ovEuEUxXQ

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en

WEBVTT Kind: captions Language: en 00:00:01.240 --> 00:00:04.650 Welcome to my presentation on equivalent fractions. 00:00:04.650 --> 00:00:07.000 So equivalent fractions are, essentially what 00:00:07.000 --> 00:00:07.560 they sound like. 00:00:07.560 --> 00:00:09.640 They're two fractions that although they use different 00:00:09.640 --> 00:00:12.260 numbers, they actually represent the same thing. 00:00:12.260 --> 00:00:13.930 Let me show you an example. 00:00:13.930 --> 00:00:18.306 Let's say I had the fraction 1/2. 00:00:18.306 --> 00:00:21.060 Why isn't it writing. 00:00:21.060 --> 00:00:23.480 Let me make sure I get the right color here. 00:00:23.480 --> 00:00:26.930 Let's say I had the fraction 1/2. 00:00:26.930 --> 00:00:30.960 So graphically, if we to draw that, if I had a pie and I 00:00:30.960 --> 00:00:33.030 would have cut it into two pieces. 00:00:33.030 --> 00:00:34.590 That's the denominator there, 2. 00:00:34.590 --> 00:00:38.420 And then if I were to eat 1 of the 2 pieces I would 00:00:38.420 --> 00:00:41.240 have eaten 1/2 of this pie. 00:00:41.240 --> 00:00:42.150 Makes sense. 00:00:42.150 --> 00:00:44.170 Nothing too complicated there. 00:00:44.170 --> 00:00:45.900 Well, what if instead of dividing the pie into two 00:00:45.900 --> 00:00:50.040 pieces, let me just draw that same pie again. 00:00:50.040 --> 00:00:52.030 Instead of dividing it in two pieces, what if I divided 00:00:52.030 --> 00:00:55.200 that pie into 4 pieces? 00:00:55.200 --> 00:00:58.870 So here in the denominator I have a possibility of-- total 00:00:58.870 --> 00:01:02.680 of 4 pieces in the pie. 00:01:02.680 --> 00:01:05.030 And instead of eating one piece, this time I actually 00:01:05.030 --> 00:01:07.135 ate 2 of the 4 pieces. 00:01:13.140 --> 00:01:15.450 Or I ate 2/4 of the pie. 00:01:15.450 --> 00:01:20.240 Well if we look at these two pictures, we can see that 00:01:20.240 --> 00:01:22.230 I've eaten the same amount of the pie. 00:01:22.230 --> 00:01:24.780 So these fractions are the same thing. 00:01:24.780 --> 00:01:28.170 If someone told you that they ate 1/2 of a pie or if they 00:01:28.170 --> 00:01:31.410 told you that they ate 2/4 of a pie, it turns out of that they 00:01:31.410 --> 00:01:32.640 ate the same amount of pie. 00:01:32.640 --> 00:01:34.490 So that's why we're saying those two fractions 00:01:34.490 --> 00:01:35.460 are equivalent. 00:01:35.460 --> 00:01:38.820 Another way, if we actually had-- let's do another one. 00:01:38.820 --> 00:01:43.770 Let's say-- and that pie is quite ugly, but let's assume 00:01:43.770 --> 00:01:45.550 it's the same type of pie. 00:01:45.550 --> 00:01:51.250 Let's say we divided that pie into 8 pieces. 00:01:51.250 --> 00:01:57.840 And now, instead of eating 2 we ate 4 of those 8 pieces. 00:01:57.840 --> 00:02:00.360 So we ate 4 out of 8 pieces. 00:02:00.360 --> 00:02:03.140 Well, we still ended up eating the same amount of the pie. 00:02:03.140 --> 00:02:05.080 We ate half of the pie. 00:02:05.080 --> 00:02:10.790 So we see that 1/2 will equal 2/4, and that equals 4/8. 00:02:10.790 --> 00:02:13.320 Now do you see a pattern here if we just look at the 00:02:13.320 --> 00:02:18.540 numerical relationships between 1/2, 2/4, and 4/8? 00:02:18.540 --> 00:02:24.640 Well, to go from 1/2 to 2/4 we multiply the denominator-- the 00:02:24.640 --> 00:02:27.310 denominator just as review is the number on the bottom 00:02:27.310 --> 00:02:29.170 of the fraction. 00:02:29.170 --> 00:02:31.020 We multiply the denominator by 2. 00:02:31.020 --> 00:02:35.370 And when you multiply the denominator by 2, we also 00:02:35.370 --> 00:02:38.250 multiply the numerator by 2. 00:02:38.250 --> 00:02:39.360 We did the same thing here. 00:02:42.360 --> 00:02:46.540 And that makes sense because well, if I double the number of 00:02:46.540 --> 00:02:50.640 pieces in the pie, then I have to eat twice as many pieces to 00:02:50.640 --> 00:02:53.700 eat the same amount of pie. 00:02:53.700 --> 00:02:56.390 Let's do some more examples of equivalent fractions 00:02:56.390 --> 00:03:00.740 and hopefully it'll hit the point home. 00:03:00.740 --> 00:03:02.020 Let me erase this. 00:03:06.550 --> 00:03:07.645 Why isn't it letting me erase? 00:03:14.030 --> 00:03:16.470 Let me use the regular mouse. 00:03:16.470 --> 00:03:17.620 OK, good. 00:03:17.620 --> 00:03:18.750 Sorry for that. 00:03:18.750 --> 00:03:20.985 So let's say I had the fraction 3/5. 00:03:24.160 --> 00:03:26.850 Well, by the same principle, as long as we multiply the 00:03:26.850 --> 00:03:31.250 numerator and the denominator by the same numbers, we'll 00:03:31.250 --> 00:03:32.740 get an equivalent fraction. 00:03:32.740 --> 00:03:38.230 So if we multiply the numerator times 7 and the denominator 00:03:38.230 --> 00:03:46.820 times 7, we'll get 21-- because 3 times 7 is 21-- over 35. 00:03:46.820 --> 00:03:51.790 And so 3/5 and 21/35 are equivalent fractions. 00:03:51.790 --> 00:03:54.880 And we essentially, and I don't know if you already know how to 00:03:54.880 --> 00:03:57.830 multiply fractions, but all we did is we multiplied 3/5 00:03:57.830 --> 00:04:02.460 times 7/7 to get 21/35. 00:04:02.460 --> 00:04:06.480 And if you look at this, what we're doing here isn't magic. 00:04:06.480 --> 00:04:09.090 7/7, well what's 7/7? 00:04:09.090 --> 00:04:12.700 If I had 7 pieces in a pie and I were to eat 7 of 00:04:12.700 --> 00:04:14.850 them; I ate the whole pie. 00:04:14.850 --> 00:04:19.160 So 7/7, this is the same thing as 1. 00:04:19.160 --> 00:04:22.620 So all we've essentially said is well, 3/5 and we 00:04:22.620 --> 00:04:23.970 multiplied it times 1. 00:04:26.910 --> 00:04:30.470 Which is the same thing as 7/7. 00:04:30.470 --> 00:04:33.490 Oh boy, this thing is messing up. 00:04:33.490 --> 00:04:38.660 And that's how we got 21/35. 00:04:38.660 --> 00:04:39.180 So it's interesting. 00:04:39.180 --> 00:04:41.090 All we did is multiply the number by 1 and we know 00:04:41.090 --> 00:04:43.690 that any number times 1 is still that number. 00:04:43.690 --> 00:04:45.940 And all we did is we figured out a different way 00:04:45.940 --> 00:04:54.150 of writing 21/35. 00:04:54.150 --> 00:04:59.930 Let's start with a fraction 5/12. 00:04:59.930 --> 00:05:05.070 And I wanted to write that with the denominator-- let's say I 00:05:05.070 --> 00:05:09.290 wanted to write that with the denominator 36. 00:05:09.290 --> 00:05:13.050 Well, to go from 12 to 36, what do we have to multiply by? 00:05:13.050 --> 00:05:17.530 Well 12 goes into 36 three times. 00:05:17.530 --> 00:05:19.830 So if we multiply the denominator by 3, we also have 00:05:19.830 --> 00:05:22.450 to multiply the numerator by 3. 00:05:22.450 --> 00:05:24.215 Times 3. 00:05:24.215 --> 00:05:27.080 We get 15. 00:05:27.080 --> 00:05:31.890 So we get 15/36 is the same thing as 5/12. 00:05:31.890 --> 00:05:34.380 And just going to our original example, all that's saying 00:05:34.380 --> 00:05:38.300 is, if I had a pie with 12 pieces and I ate 5 of them. 00:05:38.300 --> 00:05:39.140 Let's say I did that. 00:05:39.140 --> 00:05:41.990 And then you had a pie, the same size pie, you had a 00:05:41.990 --> 00:05:44.740 pie with 36 pieces and you ate 15 of them. 00:05:44.740 --> 00:05:47.540 Then we actually ate the same amount of pie.

Greatest common factor explained

https://www.youtube.com/watch?v=jFd-6EPfnec

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https://www.youtube.com/api/timedtext?v=jFd-6EPfnec&ei=hmeUZYKZEey4mLAPxo-AqAU&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249846&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=4C065F2EB4CCA96D6D4742389DFB6E51550C729F.7452636A9A5B99674C3F5E7440AC77AFDD058F19&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.180 --> 00:00:03.820 Welcome to the greatest common divisor or greatest 00:00:03.820 --> 00:00:06.030 common factor video. 00:00:06.030 --> 00:00:08.850 So just to be clear, first of all, when someone asks you 00:00:08.850 --> 00:00:16.530 whether what's the greatest common divisor of 12 and 8? 00:00:16.530 --> 00:00:19.060 Or they ask you what's the greatest common 00:00:19.060 --> 00:00:22.790 factor of 12 and 8? 00:00:22.790 --> 00:00:25.150 That's a c right there for common. 00:00:25.150 --> 00:00:26.600 I don't know why it came out like that. 00:00:26.600 --> 00:00:27.640 They're asking you the same thing. 00:00:27.640 --> 00:00:30.260 I mean, really a divisor is just a number that can divide 00:00:30.260 --> 00:00:33.100 into something, and a factor-- well, I think, that's also a 00:00:33.100 --> 00:00:34.190 number that can divide into something. 00:00:34.190 --> 00:00:37.140 So a divisor and a factor are kind of the same thing. 00:00:37.140 --> 00:00:40.200 So with that out of the way, let's figure out, what is the 00:00:40.200 --> 00:00:42.420 greatest common divisor or the greatest common 00:00:42.420 --> 00:00:43.900 factor of 12 and 8? 00:00:43.900 --> 00:00:45.980 Well, what we do is, it's pretty straightforward. 00:00:45.980 --> 00:00:48.940 First we just figure out the factors of each of the numbers. 00:00:48.940 --> 00:00:52.370 So first let's write all of the factors out of the number 12. 00:00:52.370 --> 00:00:57.260 Well, 1 is a factor, 2 goes into 12. 00:00:57.260 --> 00:00:59.100 3 goes into 12. 00:00:59.100 --> 00:01:00.780 4 goes into 12. 00:01:00.780 --> 00:01:03.960 5 does not to go into 12. 00:01:03.960 --> 00:01:06.710 6 goes into 12 because 2 times 6. 00:01:06.710 --> 00:01:10.230 And then, 12 goes into 12 of course. 00:01:10.230 --> 00:01:11.080 1 times 12. 00:01:11.080 --> 00:01:12.940 So that's the factors of 12. 00:01:12.940 --> 00:01:15.440 Let's write the factors of 8. 00:01:15.440 --> 00:01:17.610 Well, 1 goes into 8. 00:01:17.610 --> 00:01:18.950 2 goes into 8. 00:01:18.950 --> 00:01:20.640 3 does not go into 8. 00:01:20.640 --> 00:01:22.930 4 does go into 8. 00:01:22.930 --> 00:01:27.840 And then the last factor, pairing up with the 1 is 8. 00:01:27.840 --> 00:01:31.100 So now we've written all the factors of 12 and 8. 00:01:31.100 --> 00:01:34.580 So let's figure out what the common factors of 12 and 8 are. 00:01:34.580 --> 00:01:37.020 Well, they both have the common factor of 1. 00:01:37.020 --> 00:01:38.390 And that's really not so special. 00:01:38.390 --> 00:01:41.370 Pretty much every whole number or every integer has 00:01:41.370 --> 00:01:43.500 the common factor of 1. 00:01:43.500 --> 00:01:47.930 They both share the common factor 2 and they both 00:01:47.930 --> 00:01:51.080 share the common factor 4. 00:01:51.080 --> 00:01:54.620 So we're not just interested in finding a common factor, we're 00:01:54.620 --> 00:01:57.390 interested in finding the greatest common factor. 00:01:57.390 --> 00:02:00.210 So all the common factors are 1, 2 and 4. 00:02:00.210 --> 00:02:01.590 And what's the greatest of them? 00:02:01.590 --> 00:02:02.830 Well, that's pretty easy. 00:02:02.830 --> 00:02:03.900 It's 4. 00:02:03.900 --> 00:02:07.240 So the greatest common factor of 12 and 8 is 4. 00:02:07.240 --> 00:02:09.540 Let me write that down just for emphasis. 00:02:09.540 --> 00:02:14.670 Greatest common factor of 12 and 8 equals 4. 00:02:14.670 --> 00:02:16.950 And of course, we could have just as easily had said, the 00:02:16.950 --> 00:02:23.510 greatest common divisor of 12 and 8 equals 4. 00:02:23.510 --> 00:02:27.195 Sometimes it does things a little funny. 00:02:27.195 --> 00:02:28.365 Let's do another problem. 00:02:30.980 --> 00:02:41.900 What is the greatest common divisor of 25 and 20? 00:02:41.900 --> 00:02:43.775 Well, let's do it the same way. 00:02:43.775 --> 00:02:47.060 The factors of 25? 00:02:47.060 --> 00:02:48.430 Well, it's 1. 00:02:48.430 --> 00:02:49.400 2 doesn't go into it. 00:02:49.400 --> 00:02:50.160 3 doesn't go into it. 00:02:50.160 --> 00:02:51.430 4 doesn't go into it. 00:02:51.430 --> 00:02:52.065 5 does. 00:02:52.065 --> 00:02:54.280 It's actually 5 times 5. 00:02:54.280 --> 00:02:57.140 And then 25. 00:02:57.140 --> 00:02:59.570 It's interesting that this only has 3 factors. 00:02:59.570 --> 00:03:01.870 I'll leave you to think about why this number only has 3 00:03:01.870 --> 00:03:04.620 factors and other numbers tend to have an even 00:03:04.620 --> 00:03:05.245 number of factors. 00:03:07.940 --> 00:03:09.525 And then now we do the factors of 20. 00:03:12.670 --> 00:03:21.020 Factors of 20 are 1, 2, 4, 5, 10, and 20. 00:03:21.020 --> 00:03:23.300 And if we just look at this by inspection we see, well, they 00:03:23.300 --> 00:03:25.060 both share 1, but that's nothing special. 00:03:25.060 --> 00:03:28.110 But they both have the common factor of? 00:03:28.110 --> 00:03:30.560 You got it-- 5. 00:03:30.560 --> 00:03:37.300 So the greatest common divisor or greatest common factor of 25 00:03:37.300 --> 00:03:41.040 and 20- well, that equals 5. 00:03:41.040 --> 00:03:42.120 Let's do another problem. 00:03:44.950 --> 00:03:54.700 What is the greatest common factor of 5 and 12? 00:03:54.700 --> 00:03:56.430 Well, factors of 5? 00:03:56.430 --> 00:03:57.330 Pretty easy. 00:03:57.330 --> 00:03:59.350 1 and 5. 00:03:59.350 --> 00:04:00.360 That's because it's a prime number. 00:04:00.360 --> 00:04:03.080 It has no factors other than 1 and itself. 00:04:03.080 --> 00:04:05.380 Then the factors of 12? 00:04:05.380 --> 00:04:06.180 12 has a lot of factors. 00:04:06.180 --> 00:04:14.270 It's 1, 2, 3, 4, 6, and 12. 00:04:14.270 --> 00:04:20.530 So it really looks like only common factor they share is 1. 00:04:20.530 --> 00:04:23.370 So that was, I guess, in some ways kind of disappointing. 00:04:23.370 --> 00:04:28.760 So the greatest common factor of 5 and 12 is 1. 00:04:28.760 --> 00:04:31.510 And I'll throw out some terminology here for you. 00:04:31.510 --> 00:04:34.040 When two numbers have a greatest common factor of 00:04:34.040 --> 00:04:37.210 only 1, they're called relatively prime. 00:04:37.210 --> 00:04:40.100 And that kind of makes sense because a prime number is 00:04:40.100 --> 00:04:42.890 something that only has 1 and itself as a factor. 00:04:42.890 --> 00:04:45.730 And two relatively prime numbers are numbers that 00:04:45.730 --> 00:04:50.200 only have 1 as their greatest common factor. 00:04:50.200 --> 00:04:51.690 Hope I didn't confuse you. 00:04:51.690 --> 00:04:52.600 Let's do another problem. 00:04:56.780 --> 00:05:04.580 Let's do the greatest common divisor of 6 and 12. 00:05:04.580 --> 00:05:05.680 I know 12's coming up a lot. 00:05:05.680 --> 00:05:08.820 I'll try to be more creative when I think of my numbers. 00:05:08.820 --> 00:05:11.050 Well, the greatest common divisor of 6 and 12? 00:05:11.050 --> 00:05:12.920 Well, it's the factors of 6. 00:05:12.920 --> 00:05:17.770 Are 1, 2, 3, and 6. 00:05:17.770 --> 00:05:23.480 Factors of 12: 1, 2, 3-- we should have these 00:05:23.480 --> 00:05:23.980 memorized by now. 00:05:23.980 --> 00:05:29.000 3, 4, 6, and 12. 00:05:29.000 --> 00:05:33.730 Well, it turns out 1 is a common factor of both. 00:05:33.730 --> 00:05:36.350 2 is also a common factor of both. 00:05:36.350 --> 00:05:39.550 3 is a common factor of both. 00:05:39.550 --> 00:05:42.100 And 6 is a common factor of both. 00:05:42.100 --> 00:05:43.920 And of course, what's the greatest common factor? 00:05:43.920 --> 00:05:45.540 Well, it's 6. 00:05:45.540 --> 00:05:46.780 And that's interesting. 00:05:46.780 --> 00:05:49.610 So in this situation the greatest common divisor-- and I 00:05:49.610 --> 00:05:52.600 apologize that I keep switching between divisor and factor. 00:05:52.600 --> 00:05:54.160 The mathematics community should settle on 00:05:54.160 --> 00:05:55.200 one of the two. 00:05:55.200 --> 00:06:00.220 The greatest common divisor of 6 and 12 equals 6. 00:06:00.220 --> 00:06:01.680 So it actually equals one of the numbers. 00:06:01.680 --> 00:06:04.270 And that makes a lot of sense because 6 actually 00:06:04.270 --> 00:06:07.720 is divisible into 12. 00:06:07.720 --> 00:06:08.940 Well, that's it for now. 00:06:08.940 --> 00:06:11.650 Hopefully you're ready to do the greatest common divisor 00:06:11.650 --> 00:06:12.820 or factor problems. 00:06:12.820 --> 00:06:14.960 I think I might make another module in the near future 00:06:14.960 --> 00:06:17.550 that'll give you more example problems.

Multiplication 7: Old video giving more examples

https://www.youtube.com/watch?v=_k3aWF6_b4w

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https://www.youtube.com/api/timedtext?v=_k3aWF6_b4w&ei=hmeUZZHgEOW_mLAP4Yuf6Ag&caps=asr&opi=112496729&xoaf=5&hl=en&ip=0.0.0.0&ipbits=0&expire=1704249846&sparams=ip%2Cipbits%2Cexpire%2Cv%2Cei%2Ccaps%2Copi%2Cxoaf&signature=07B1ABB85F0B4B9C8026F0CD73F37D7BA18A477E.548BBD74FC58C74802FB1069B86EE1996756C9EE&key=yt8&lang=en&fmt=vtt

en

WEBVTT Kind: captions Language: en 00:00:01.290 --> 00:00:02.880 Welcome to Level 4 multiplication. 00:00:02.880 --> 00:00:04.415 Let's do some problems. 00:00:04.415 --> 00:00:11.580 Let's see, we had 235 times-- I'm going to use two different 00:00:11.580 --> 00:00:14.330 colors here, so bear with me a second. 00:00:14.330 --> 00:00:16.295 Let's say times 47. 00:00:21.720 --> 00:00:24.210 So you start a Level 4 problem just like you would normally 00:00:24.210 --> 00:00:26.960 do a Level 3 problem. 00:00:26.960 --> 00:00:31.070 We'll take that 7, and we'll multiply it by 235. 00:00:31.070 --> 00:00:33.350 So 7 times 5 is 35. 00:00:36.140 --> 00:00:41.640 7 times 3 is 21, plus the 3 we just carried is 24. 00:00:44.660 --> 00:00:49.790 7 times 2 is 14, plus the 2 we just carried. 00:00:49.790 --> 00:00:51.050 This is 16. 00:00:51.050 --> 00:00:52.570 So we're done with the 7. 00:00:52.570 --> 00:00:54.400 Now we have to deal with this 4. 00:00:54.400 --> 00:00:58.250 Well, since that 4 is in the tens place, we add a 0 here. 00:00:58.250 --> 00:01:01.710 You could almost view it as we're multiplying 235, not by 00:01:01.710 --> 00:01:04.300 4, but we're multiplying it by 40, and that's why 00:01:04.300 --> 00:01:06.500 we put that 0 there. 00:01:06.500 --> 00:01:09.320 But once you put the 0 there, you can treat it just like a 4. 00:01:09.320 --> 00:01:11.990 So you say 4 times 5, well, that's 20. 00:01:14.990 --> 00:01:16.850 Let's ignore what we had from before. 00:01:16.850 --> 00:01:24.190 4 times 3 is 12, plus the 2 we just carried, which is 14. 00:01:24.190 --> 00:01:29.820 4 times 2 is 8, plus the 1 we just carried, so that's 9. 00:01:29.820 --> 00:01:32.010 And now we just add up everything. 00:01:32.010 --> 00:01:40.580 5 plus 0 is 5, 4 plus 0 is 4, 6 plus 4 is 10, carry the 1, and 00:01:40.580 --> 00:01:43.680 1 plus 9, well, that's 11. 00:01:43.680 --> 00:01:46.230 So the answer's 11,045. 00:01:46.230 --> 00:01:48.160 Let's do another problem. 00:01:48.160 --> 00:01:57.830 Let's say I had 873 times-- and I'm making these numbers up on 00:01:57.830 --> 00:02:05.800 the fly, so bear with me-- 873 times-- some high numbers-- 00:02:05.800 --> 00:02:09.230 90-- and I'm doing them in different colors, just so you 00:02:09.230 --> 00:02:10.970 hopefully get a better understanding of what 00:02:10.970 --> 00:02:13.130 I'm trying to explain. 00:02:13.130 --> 00:02:14.595 Let's say 97. 00:02:14.595 --> 00:02:15.960 No, I just used a 7. 00:02:15.960 --> 00:02:16.705 Let's make it 98. 00:02:19.880 --> 00:02:22.660 So just like we did before, we go to the ones place first, and 00:02:22.660 --> 00:02:26.660 that's where that 8 is, and we multiply that 8 times 873. 00:02:26.660 --> 00:02:32.080 So 8 times 3 is 24, carry the 2. 00:02:32.080 --> 00:02:40.100 8 times 7 is 56, plus 2 is 58, carry the 5. 00:02:40.100 --> 00:02:44.420 8 times 8 is 64, plus the 5 we just carried. 00:02:44.420 --> 00:02:46.480 That's 69. 00:02:46.480 --> 00:02:47.580 We're done with the 8. 00:02:47.580 --> 00:02:50.220 Now we have to multiply the 9, or we could just do it as 00:02:50.220 --> 00:02:53.840 we're multiplying 873 by 90. 00:02:53.840 --> 00:02:56.170 But multiplying something by 90 is just the same thing as 00:02:56.170 --> 00:02:59.320 multiplying something by 9 and then adding a 0 at the end, so 00:02:59.320 --> 00:03:01.505 that's why I put a 0 here. 00:03:01.505 --> 00:03:04.740 Let's say 9 times 3-- well, first, just to clean up 00:03:04.740 --> 00:03:06.720 things, let's get rid of what we had from before. 00:03:06.720 --> 00:03:12.950 We say 9 times 3 is 27, carry the 2. 00:03:12.950 --> 00:03:17.600 9 times 7 is 63, plus the 2 that we just carried is 65. 00:03:20.850 --> 00:03:22.350 Carry the 6. 00:03:22.350 --> 00:03:25.490 8 times 8 is 72, plus the 6 we just carried. 00:03:25.490 --> 00:03:28.040 That's 78. 00:03:28.040 --> 00:03:29.720 And now we just add again. 00:03:29.720 --> 00:03:37.680 4, 8 plus 7 is 15, 1 plus 9 plus 5 is 15, 1 plus 6 plus 00:03:37.680 --> 00:03:42.170 8 is also 15, and 1 plus 7, that's 8. 00:03:42.170 --> 00:03:44.470 So the answer, hopefully-- I don't have a calculator in 00:03:44.470 --> 00:03:48.760 front of me-- is 85,554, assuming I didn't make 00:03:48.760 --> 00:03:50.890 any careless mistakes. 00:03:50.890 --> 00:03:51.910 Let's do one more problem. 00:03:51.910 --> 00:03:54.670 I think it'll hit the point home. 00:03:54.670 --> 00:03:56.530 The next problem I'm going to do, you can almost do it as a 00:03:56.530 --> 00:03:59.630 Level 5 problem because I'm actually going to multiply two 00:03:59.630 --> 00:04:02.180 three-digit numbers, but it's really the same thing, and 00:04:02.180 --> 00:04:03.770 hopefully, you'll see the pattern. 00:04:03.770 --> 00:04:11.740 So let's say I had 234 times-- and I'm going to use three 00:04:11.740 --> 00:04:23.150 colors now-- let's say 643. 00:04:23.150 --> 00:04:26.030 So first we do the 3, which is in the ones place, and 00:04:26.030 --> 00:04:28.950 we multiply that times 234. 00:04:28.950 --> 00:04:32.310 Well, 3 times 4 is 12, carry the 1. 00:04:32.310 --> 00:04:36.570 3 times 3 is 9, add the 1. 00:04:36.570 --> 00:04:38.620 That's 10, carry the 1. 00:04:38.620 --> 00:04:40.880 3 times 2 is 6, plus 1. 00:04:40.880 --> 00:04:43.730 Well, that's 7. 00:04:43.730 --> 00:04:45.970 And then we've done the-- I think I've made a 00:04:45.970 --> 00:04:46.960 mistake someplace. 00:04:46.960 --> 00:04:48.440 Let me see. 00:04:48.440 --> 00:04:50.880 3 times 4 is 12. 00:04:50.880 --> 00:04:54.020 Oh, no, I think that's correct. 00:04:54.020 --> 00:04:56.440 I was confusing myself. 00:04:56.440 --> 00:04:59.020 OK, now we're ready to do the 4, or the 40, and since it's a 00:04:59.020 --> 00:05:03.010 40 because it's in the tens place, we put a 0 right here. 00:05:03.010 --> 00:05:06.000 We say 4 times 4-- well, let's clean up this stuff at the top. 00:05:06.000 --> 00:05:08.310 I always forget to do that. 00:05:08.310 --> 00:05:12.110 4 times 4, well, that's 16, carry the 1. 00:05:12.110 --> 00:05:19.130 4 times 3, well, that's 12, plus 1, well, that's 13. 00:05:19.130 --> 00:05:23.680 4 times 2 is 8, plus the 1, well, that's 9. 00:05:23.680 --> 00:05:25.940 And now we're done with the 4, or the 40, depending on 00:05:25.940 --> 00:05:26.520 how you want to view it. 00:05:26.520 --> 00:05:28.810 Now we're ready for the 6, or the 600. 00:05:28.810 --> 00:05:33.120 Since it's a 600, we put two zeroes here, and we 00:05:33.120 --> 00:05:34.170 just treat it like a 6. 00:05:34.170 --> 00:05:37.110 Let's clean up what we did before. 00:05:37.110 --> 00:05:41.220 So 6 times 4 is 24, carry the 2. 00:05:41.220 --> 00:05:47.500 6 times 3 is 18, plus 2 is 20, carry the 2. 00:05:47.500 --> 00:05:52.710 6 times 2 is 12, plus 2 is 14. 00:05:52.710 --> 00:05:55.470 And now we add it all up. 00:05:55.470 --> 00:05:59.680 2, 6, 7 plus 3 is 10. 00:05:59.680 --> 00:06:02.140 14, carry the 1. 00:06:02.140 --> 00:06:05.000 1 plus 9 is 10, carry the 1. 00:06:05.000 --> 00:06:06.780 That's 5. 00:06:06.780 --> 00:06:07.090 That's 1. 00:06:07.090 --> 00:06:07.680 I hope you can see. 00:06:07.680 --> 00:06:09.870 I hope it's not falling off the screen, the answer 00:06:09.870 --> 00:06:13.700 I get is a 150,462. 00:06:13.700 --> 00:06:15.660 I think you're ready now to try Level 4 00:06:15.660 --> 00:06:17.640 multiplication problems.