In this video we're going to find the intersection point of two lines and we're going to do it without graphing the lines. Now one way to do this to get the intersection would be simply to graph the lines. You'd plot a graph and the slope of this line is four. The y intercept is negative one. And then you'd do the slope of the y intercept of the other one and figure out what the intersection point is. That's all fine and good. The reason for not doing it that way is that it's not very precise. Sometimes if your intersection point is 2.14 you might look at that and say it's the point 2.4 because it looks like it on the graph. But it's not. So doing it algebraically gets us a much more precise answer to our intersection point. So what do we do? Well here we have two lines. Let's remember exactly what the equation of lines means. This line, every point on the line, the y coordinate is four times the x coordinate minus one. That's what that equation means. It means that for every single point on that line, the y coordinate at that point is four times the x. And for every single point on this line, the y coordinate is negative x coordinate plus 19. Now there's one point where these two lines cross each other. Where the y coordinate is both 4x minus 1 and negative x plus 19. So for that one point, if the y coordinate equals this and the y coordinate also equals this, then these two things have to equal each other. So what we do is we set up 4x minus 1 equals negative x plus 19. Have one equation equal the other equation and just solve that. Not so hard. A lot of x is up there. Let me add x to this side. Add x to this side. And now I've got 5x minus 1 equals 19. I'm going to add 1 to both sides here. And I get 5 times x equals 20, which means x must equal 4. Great. Okay. Am I done? No. I'm looking for a point. And all I have is an x. I still need a y coordinate. So it's going to be 4. Oops. It's going to be 4 something. How do I find the y coordinate? I take my 4 and I plug it right into either one of the equations. My preference is both equations. Because if you can put it into both equations and get the same y coordinate, you know you're right. Okay. So 4 times 4 is 16 minus 1 is 15. So that must be our mean. Our answer is the point 415. Maybe. Let's see. Negative 4 plus 19 is also 15. So yes. Now I know. It's exactly right. Okay. Not so hard. Is it? Let's try another problem. Let's do. Y equals 1 half x minus 7 and y equals x minus 11. Okay. We're going to do the exact same thing. We're going to say this equation is going to equal this equation. So I get 1 half x minus 7 equals x minus 11. Again, a lot of x is up there. Let's give it up some of them. Let's subtract 1 half x from both sides. And I get these cancel out. Negative 7 equals x minus half of x. That's like saying 1 minus 1 half. That's 1 half x minus 11. Now I'm going to add 11 to both sides. And I get negative 7 plus 11 is 4 equals 1 half x. So 1 half of what number it was for. I believe x would be 8. Okay. Dividing by 1 half is the same thing as multiplying by 2. So I get x equals 8. Great. That means my point is 8 something. Half of 8 is 4 minus 7 is negative 3. And 8 minus 11 is also negative 3. So again, it works. Okay. Let me give you one more. And this one is going to have a little twist to it. This time we have y equals negative 2x plus 2. And 3x plus 2y equals 1. Now you've already noticed what's different about this one. And that is this line right here is not written in slope intercept form. This one is. It's solved for y. Y equals mx. My slope is negative 2. My y intercept is the point 0 to. This line's not written right though. So one option would be you subtract 3x from both sides. You divide both sides by 2. You solve for y. You figure out what the... And then you set this equation equal to that equation. That's a perfectly fine way to do it. Actually, let me take that back. That's an okay way to do it. I don't recommend doing it that way. It ends up being the hard way. Let me show you a much easier way. Y is negative 2x plus 2, right? Y equals that. That means y and negative 2x plus 2 are exactly the same thing. If one thing is exactly the same thing as another thing, those things should be interchangeable. So what I can do is I can take this... And let me just replace this y here with negative 2x plus 2. Because after all, they are the same thing. All right. Let's just make that little substitution. We've got 3x plus 2 times... And I'm not going to call it y this time. I'm going to call it negative 2x plus 2. And that equals... finish it off here. One. Okay? Let me show you again exactly what we're doing here. We're just rewriting this line. Except instead of saying y, I'm saying negative 2x plus 2. And why am I doing that? Because y equals negative 2x plus 2. Because it's the same thing. All right. We know how to solve this. Distribute, combine like terms. Let's go ahead and do it. It's pretty easy. 3x plus 2 times negative 2x is negative 4x. 2 times 2 is 4 equals 1. 3x plus negative 4x. That gets me on negative x plus 4 equals 1. Subtract to the 4. I get negative x equals negative 3. Multiplying times negative 1, I get x equals 3. Okay? And if x is 3, let me pop that back up here. And negative 2x3 would be negative 6 plus 2 is negative 4. So it looks like my answer is 3 negative 4. Let me just take this point, which I know works for this one. And see if it works for this line. 3 times 3 is 9 plus 2 times negative 4 is negative 8. And 9 plus negative 8 does equal 1. So this point is on this line. And that means it's the intersection point. Okay? This method here, actually, the method we've been using this entire time is called the substitution method. And the reason it's the substitution method is we're taking what y equals and we're substituting y for that expression instead.