INPUT ARTICLE: Article: The substitution method is ideal when one of the coefficients in one of the equations is equal to one. Then, all you have to do is isolate the single-coefficient variable on one side of the equation to find its value. If you're working with the equations 2x + 3y = 9 and x + 4y = 2, you should isolate x in the second equation. x + 4y = 2 x = 2 - 4y Take the value you found when you isolated the variable and replace that value instead of the variable in the equation that you did not manipulate. You won't be able to solve anything if you plug it back into the equation you just manipulated. Here's what to do: x = 2 - 4y --> 2x + 3y = 9 2(2 - 4y) + 3y = 9 4 - 8y + 3y = 9 4 - 5y = 9 -5y = 9 - 4 -5y = 5 -y = 1 y = - 1 Now that you know that y = - 1, just plug that value into the simpler equation to find the value of x. Here's how you do it: y = -1 --> x = 2 - 4y x = 2 - 4(-1) x = 2 - -4 x = 2 + 4 x = 6 You have solved the system of equations by substitution. (x, y) = (6, -1) To make sure that you solved the system of equations correctly, you can just plug in your two answers to both equations to make sure that they work both times. Here's how to do it: Plug (6, -1) in for (x, y) in the equation 2x + 3y = 9. 2(6) + 3(-1) = 9 12 - 3 = 9 9 = 9 Plug (6, -1) in for (x, y) in the equation x + 4y = 2. 6 + 4(-1) = 2 6 - 4 = 2 2 = 2

SUMMARY: Isolate one variable. Plug the value of the variable you isolated back into the other equation. Solve for the remaining variable. Check your work.

INPUT ARTICLE: Article: Look at the inside of a ring's band to find the hallmark that indicates the platinum purity. Find this marking as a sure way to determine that the ring is truly made of platinum. You may need a magnifying glass to see the fine engraving on a small ring. If there is no engraved hallmark, the ring is not platinum, or is made with less than 50% of it and cannot be considered and marked as platinum. Check the inner band for a symbol that you can use to quickly identify if the ring is made with real platinum. The type and presence of the symbol depends on the country of origin; sometimes there may only be words and numbers instead of a symbol. In the U.S., you will see a symbol that resembles a house shape (triangle atop a rectangle), or simply a combination of letters and numbers. In India, you will find a symbol created by an uppercase "P" and a lowercase "t," followed by the number 950. In China, you will find the English letters "Pt," perhaps alongside other Chinese characters. In Japan, you will see "Pt" followed by the number 850, 900, 950, or 999. Look at the numbers and/or letters that are given as part of the platinum hallmark. The letters will designate the ring as platinum, while the numbers indicate how many parts out of 1000 are pure platinum by weight. If letters are given, they will say either PT, PLAT, or PLATINUM, which indicates that the ring is at least 95% platinum. In the U.S., the number given before or after these letters can be 850, 900, or 950. This indicates that the ring is made with 85%, 90%, or 95% pure platinum, respectively. The other percentage of the ring composition is a metal that the platinum is alloyed with to make it harder.

SUMMARY: Locate the hallmark on the inside of the band. Look for a symbol that indicates platinum. Read the letters and numbers of the hallmark.

INPUT ARTICLE: Article: If you have lost 4 pounds or fewer, then you may want to consider staying in Induction longer. If you are still having significant struggles with carbohydrate cravings, then you may want to continue Induction until those cravings are eliminated. If you have high blood pressure, Type 2 diabetes, sleep apnea or a BMI of over 30, then you should consider staying on Induction for more rapid weight loss. Always follow the guidance of your doctor. This will provide better nutrition and more variety. However, you should continue to eat less than 20 grams of carbohydrates each day. You'll add 5 daily carbohydrate grams each week until you notice your weight loss slowing. This will let you know how many carbohydrates you can eat while still losing weight. Perform cardiovascular exercise and strength training as soon as you leave Induction so that you can lose weight more quickly.

SUMMARY: Evaluate how much weight you have lost. Consider your cravings. Be honest about your health issues. Add 1 ounce of nuts or seeds if you decide to stay in Induction. Progress to the OWL phase when you're ready. Start exercising.

INPUT ARTICLE: Article: The cost will depend upon how long you intend to rent the car. Take the on ramp to US-101 N. Take that exit and turn right onto Market Street as you get off the freeway. Follow Market Street east until you get to your downtown destination.

SUMMARY:
Board the blue line on the Air Train, until you reach the Rental Car Center Station. Rent a car. Exit the airport heading west, turn right at the fork. Follow the US-101 North until you see the Octavia/Fell exit.