Idiot\'s Guides Basic Math and Pre-Algebra

106 Part 2: Into the Unknown

Dividing with Variables ........................................................................................................

When you try to divide by a variable or an expression that contains a variable, you have the

same problem as you did for multiplication: you don’t know what number the variable stands for.

Just as it did in multiplication, that means you can’t always do as much simplifying as you might

like to, but in division, it also causes another problem. You can multiply by any number, but you

can’t divide by zero. If you don’t know what number the variable stands for, you don’t know what

you’re dividing by, and because division by zero is undefined, that’s dangerous.

That’s why whenever you divide by a variable or write a fraction with a variable in the

denominator, you always include a note that says, in one way or another, “as long as this isn’t

zero.” If you wanted to divide 5x^2 z x, you’d include a little warning: x { 0.

Consider the Domain

When we talk about variables in algebra, we often talk about the domain of the variable. The

domain is the set of all values that you can reasonably substitute for the variable. It’s all the

values for the variable that make sense.

One of the first things you look at when you think about the domain is dividing. If you’re trying

to divide by the variable, 0 can’t be in the domain. There are other problems to be aware of as

well. You can’t take the square root of a negative number, for example, and if you’re using your

variable to solve a word problem, only numbers that make sense in the problem should be in the

domain. The length of a fence can’t be -10 feet, and you can’t invite 12.4 people to your party.

But the number one concern is making sure you never divide by zero.

Rules for Dividing with Variables

So you have to be careful and include a warning note, but how do you divide by a variable or an

expression containing a variable? There are several ways, because as the expressions get more

complicated you need different methods, but the basic rules come back to fractions and exponents.

If you have a product of constants and variables divided by a product of constants and variables,

you can think of the problem as a fraction that needs to be put in simplest form. Let’s look at

some examples.

If you want to divide 8x by 2, you can think of it as^8

2

8

21

4

11

xx x 4 x. You’re just dividing the

coefficient of the numerator by the constant in the denominator. Now it’s not hard to think of a

case, like^8

3

x, where it might be better to just leave it as it is, or y

5

where there’s nothing you can

do. But if the denominator is a constant, and you can cancel that constant with a coefficient in the

numerator, that’s what you want to do.