Variables

Definition Of Variable
What is a variable? It is a box, and it exists to contain a value. Sometimes the value is already inside the box, and you have to figure out what that value is. Other times, the box is empty, and you get to pick the value to put inside. More about that later. First...
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:

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

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In the above discussion, I illustrated both uses of variables. In the equation "x + 2 = 5", x can only have a value of 3. The statement (the equation) 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 equation "C=2πr" 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 ©
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); instead, (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.)