Solving Another Equation with Two Variables
Date: 1/30/96 at 20:28:1 From: McKellar Clan Subject: how to do algebra Dear Dr. Math, I am in the 7th grade and our math class is learning algebra. I do not understand it and my teacher is no help in trying to explain it. My mother doesn't understand algebra and my father is too busy to help me. I hope you can. I have included a sample problem from my book; 6y = 3D = 54. I am supposed to solve the equation, but I don't even understand it! PLEASE HELP!!!! I really need to get a handle on this stuff. My name is Brandon.
Date: 2/1/96 at 14:44:17 From: Doctor Elise Subject: Re: how to do algebra Hi! You sound really frustrated. Algebra makes a lot of people confused, especially at the beginning, but there's no magic to it. You really can learn it! I'll try to help by talking about algebra in general, and your sample problem in particular. Algebra is the next step in math after you can pretty much add, subtract, multiply, divide, and do anything else you want with actual numbers. In Algebra, instead of using a specific number, like '5', we start using a letter, like 'y', to represent a number we don't know yet. At this point, we also start leaving out the "times" sign when we write equations if we're multiplying a number "times" a letter. What '6y' really means is "six times y, which is a number I don't know yet." The goal of almost every algebra problem is to find out what number (or numbers) the letter could equal. The mathbooks usually call it "solving for y", and you've done it when your equation finally looks like "y = some number". Your example is: 6y = 3D = 54 This looks like a pretty funny equation, doesn't it? That's because...Surprise! It's really three entirely different problems. They are: 6y = 3D 3D = 54 6y = 54 Does that make a little more sense? Let's start with the third one. 6y = 54 In words, that's "six times y equals 54". The goal is to figure out what number we can plug in for 'y' that works. If you just think about it for a minute, you know the answer already, 6 * 9 = 54, right? Here's how you get there using algebra. The way you solve any algebra problem is by putting all the letters on one side of the equals sign, and all the numbers on the other. The big rule is that you can do anything you want to the equation as long as you do the same thing to both sides of the equals sign. Think about it. If I start with 2 = 2, I know this is true. What if I add 4 to both sides: 2 + 4 = 2 + 4 I get: 6 = 6 This is also true. What if I multiply both sides by 3: 6 * 3 = 6 * 3 I get: 18 = 18 Still works. Okay, what if I multiply both sides by some unknown number 'x': 18 * x = 18 * x This is still always true. Remember, in algebra we write this as: 18x = 18x Okay, what if I add 4 * x to both sides? Can I do that? Sure! 18x + 4x = 18x + 4x Works just fine. Now. Remember how you used to factor your plain old vanilla numbers? As in, 6 = 3 * 2? And if you wanted to add 6 + 4 you could write it like 6 + 4 3 * 2 + 2 * 2 2 * (3 + 2) 2 * 5 10 See, you get 10 this way, too. Well, you can do the same thing with this silly 'x' number. 18x + 4x 18 * x + 4 * x write it out the long way x * (18 + 4) pull out the 'x' x * 22 add the numbers 22x here's the answer. Of course, without an "equals" sign, we can't solve it any further. Anyhow, if you have "6y = 54", and what you want is "y = something", you just have to divide both sides by 6, right? 6y = 54 6y / 6 = 54 / 6 and we can write 6y/6 in a bunch of different ways, but it basically boils down to the exact same thing you used to do with fractions. The same way you can reduce 10/15 by factoring it into 5 * 2 / 5 * 3 and then cross out both 5's, we can write 6y/6 as 6 * y / 6 * 1 and cross out the 6's to get plain old y/1, which is 'y'. And, of course, 54/6 = 9, so we have y = 9 Yay! So, for 3D = 54 we do the same approach: 3D /3 = 54/3 D = 18 We can even use 6y = 3D to check our work. If you substitute 9 for y and 18 for D, is the equation true? I hope this helps. Good luck! -Doctor Elise, The Math Forum
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