Understanding Variables

Date: 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/