Understanding VariablesDate: 08/02/99 at 23:54:35 From: Kim Subject: Algebra I have always had a hard time with algebra. It makes no sense to me why someone would replace a number with a letter. Is there some secret pattern that could help me solve algebra problems? Date: 08/03/99 at 13:06:31 From: Doctor Peterson Subject: Re: Algebra Hi, Kim. You're doing something that's a little like algebra whenever you use a pronoun. You could have written this: Kim has always had a hard time with algebra. It makes no sense to Kim why someone would replace a number with a letter. Is there some secret pattern that could help Kim solve algebra problems? When you wrote us, you used the pronoun "I" or "me" to take the place of your name. A variable is like a pronoun: it's a way of talking about a number without calling it by name. We don't generally replace a number with a letter; more often we don't know the number yet, so we just give it a nickname (like "X") and work with it until we can replace the letter with the right number. The great discovery that made algebra possible was the realization that even if you don't know what a number is, you can still talk about it and know certain things about its behavior; for example, no matter what the number is, if you add 2 to it and then subtract 2 from the result, you'll have the same number you started with. We can say X + 2 - 2 = X for any X. Here's one secret that may help you: when you see an equation that confuses you, try putting an actual number in place of the variable, and see if it makes sense then. For example, in what I just wrote, you could try replacing X with 47: 47 + 2 - 2 = 47 It works! Now think about why it works: 47 plus 2 means you've gone 2 units to the right; minus 2 takes you two units back to where you started. It didn't matter that the place you started at was 47; adding 2 and subtracting 2 undo one another. Now I'll take you one step deeper into algebra and actually solve an equation. Let's say we're told that 3X - 2 = 7 In words, I can say, "I have a secret number. If I multiply it by 3 and then subtract 2, I get 7. What is it?" (Notice how I used pronouns to stand for the number.) In order to solve this, I can think of it as if the X were a present someone wrapped up for me. First they put on some "three-times" paper, and then over that they put some "-2" paper. The package I was given is a 7. I want to unwrap it and see what the X is that's inside. To take off the "subtract 2" I can add 2 (remember what we said before about adding and subtracting 2). It works like this: 3X - 2 = 7 3X - 2 + 2 = 7 + 2 3X = 9 So I've taken off the "-2" paper, and what I found inside was a 9. Now we can take off the multiplication by dividing by 3: 3X / 3 = 9 / 3 X = 3 Now the present is unwrapped, and we can see what it is. We were able to do all this because we knew how to handle a number without knowing what it was. Of course, since we can always make mistakes, we should check that we're right; let's wrap it back up and see if it's a 7: 3(3) - 2 = 9 - 2 = 7 Yup! That's what was in the package. And doing this lets us see what was happening to X, by putting a real number (the right one) in its place. I don't know whether this is the level of help you need. If you have some specific problems you'd like me to help you with, please write back and show me what you can understand and what you have trouble with. I'll be glad to help. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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