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Variables

Remember when you were in elementary school, and you were learning your addition? The teacher would hand you worksheets that said things like:

; fill in the box.

Variables are the same thing. Now we say:

x + 2 = 5 ; solve for x.

Why did we switch from boxes to letters? Because letters are better. Boxes come in only a few shapes, but letters come in many varieties, and letters can stand for something. For instance, the formula from geometry for finding a circle's circumference is:

This formula makes more sense than, say:

The two formulae say exactly the same thing, but using "C" for "circumference" and "r" for "radius" is more useful. Boxes are fine, but letters are better.

In the above discussion, I illustrated both uses of variables. In the first case, "x + 2 = 5", x can only be 3. The statement is not true for any other value. That is to say, the value of x is "fixed"; we just have to figure out what it is. On the other hand, in the second case, " ", the radius r can be any non-negative number we choose — we get to pick! — and then we get to figure out what the circumference C is. In the first case, we had to open the box to see what was already inside; in the second, we got to put the value in ourselves.

Now that we have variables, what do we do with them? Go back in your mind again to elementary school. Your teacher would have you add "2 apples plus 6 apples is 8 apples". The same rules apply to variables: "2 boxes plus 6 boxes is 8 boxes", or, using variables, "2x + 6x = 8x". "A box and another box is two boxes", or "x + x = 2x". "Two dollars, less the ten that you owe to your friend, means that you're eight dollars in the red", or "2x – 10x = –8x".

But note: "2 apples plus 6 oranges" is just 2 apples and 6 oranges; they might make a nice fruit salad, but they're not 8 of anything. In the same way, "2x + 6y" is just 2x + 6y; you can't combine the two variables into one. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

When multiplying, we use exponents. For instance, (5)(5) = 5². Of course, we can simplify this as 5² = 25. Similarly, (x)(x) = x². But, until we know what value to put in for x, we cannot simplify this.

Don't confuse multiplication and addition: (x)(x) does not equal 2x, just as (5)(5) does not equal (2)(5), so (x)(x) equals x². (Note the technique I just used: If you're not sure what to do with the variables, put in numbers, where you know what to do. Then, whatever you did with the numbers, do that with the variables.)

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