What Are Variables and How Do They Work?

Date: 06/29/2004 at 14:10:38
From: Christine
Subject: Please explain to me the meaning of alegebra

I don't understand the meaning of even basic algebra, like what 2y or
x means.  What do variables mean?

Date: 06/30/2004 at 23:35:29
From: Doctor Peterson
Subject: Re: Please explain to me the meaning of alegebra

Hi, Christine.

You're used to using numbers in arithmetic:

  2 + 3 = 5

When we define a variable, like x, it's like making a box with a label 
on it, which can hold any number:

  +---+
  |   |
  +---+
    x

We don't know what number is in the box (imagine it's wrapped in 
paper), but we know that whatever is in there is a number, so you can 
do arithmetic with it.  So we can write equations like

  +---+
  |   | + 3 = 5
  +---+
    x

This tells us that whatever is in the box (which we call the "value" 
of the variable x), plus 3, gives 5.  And if you think about it enough 
(algebra gives you easy ways to do that thinking!), you can see that 
the value HAS TO BE 2.  What you've just done is to "solve" the 
equation

  x + 3 = 5

by saying that

  x = 2

One special trick with variables is that we can write multiplication 
without using a symbol:

  2x

means 2 times x. (If we wrote 23 to mean 2 times 3, no one could tell 
that's what we meant, so we can't do that!) In this case, it we knew 
that x was 2, we would know that 2x is 2 times 2, or 4.

Here are some basic introductions to algebra that may help with these 
general ideas:

  The Why of Algebra
    http://mathforum.org/library/drmath/view/57015.html 

  What is Algebra?
    http://mathforum.org/library/drmath/view/57509.html 

  Variables: Connecting Letters and Numbers
    http://mathforum.org/library/drmath/view/57312.html 

  Letters for Variables
    http://mathforum.org/library/drmath/view/57527.html 

If you have any further questions, feel free to write back.  It will 
be best if you can show us a specific problem you don't understand, 
and tell us where you get stuck.  Then we can do more to help you out.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/