Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 9: Adding and Subtracting with Variables 111

When Can You Combine Terms?

Terms may only involve multiplication, but you need to think about what happens when you

want to do something besides multiply, specifically, when you want to add or subtract terms.

When the terms are just numbers, addition and subtraction are straightforward: 4 + 8 = 12 and

-5 + 9 = 4. You’re just adding (or subtracting) according to the rules of arithmetic. The moment

variables enter the picture, however, you’re faced with a dilemma. How do you add two numbers

if you don’t know what they are?

You may already have part of the answer. If you add x + x, whatever number x stands for, you

have two of that, so 2x. But what if you need to add x + y? You don’t have two x’s and you don’t

have 2 y’s. You can’t really say much about what you do have, except that you have x + y.

Different variables, like x and y, are unlike terms. They’re different. It’s an apple-and-orange

kind of thing. One apple plus one orange doesn’t give you two apples or two oranges or two

appleoranges. It can give you two fruits, if you take a common denominator approach to the

matter, but when you’re working with variables, you don’t know enough about them to find a

common denominator. You’re stuck admitting that unlike terms can’t be combined.

DEFINITION

Unlike terms are terms with different variables, such as x and y.

In order to combine terms, they must have the same variable. They must be like terms. As usual,

there are some complications to that simple rule. What about x + xy? Or x + x^2? In each of these

examples, the second term has an x, but it also has something else. Can you combine x with xy

(wh ich is x multiplied by y) or x^2 (wh ich is x multiplied by x)?

If you could combine x and xy, what would it give you? The xy isn’t the same number as x (unless

y happens to be 1, and you can’t bet on that) so you don’t have 2x, and you can’t just throw the y

away. The terms x and xy are unlike terms.

The x^2 may look more like the x, but it’s not the same. If x were equal to 5, x^2 would be 25, and

they’d add to 30. That’s six times the value of x. But if x were equal to 3, x^2 would be 9, and they’d

add to 12, which is four times the value of x. There are so many possibilities, and while there is a

pattern to them, it’s not immediately clear. It’s difficult to say what x + x^2 equals because x and x^2

are not like terms.