## Basic Math and Pre-Algebra

**PART 2. Into the Unknown**

**CHAPTER 9. Adding and Subtracting with Variables**

**Adding and Subtracting Like Terms**

So x + x is 2x, but x + y must stay as x + y because we have no way to combine the unlike terms. What about 2x + 3x? The variable is the same, but the coefficients, the numbers in front, are different. Is that a problem? Like terms are terms that have the same variable and the same exponent. These terms meet that rule, so why is it okay for the coefficients to be different?

Think about what the coefficients tell you. You have 2 x’s and another 3 x’s. That’s all x’s, so you can combine them. The 2x means x + x and the 3x means x + x + x. You can put them all together and end up with a total of 5 x’s. 2x + 3x = 5x.

If the variable parts of two terms are identical, you can add them by just adding the coefficients. As long as you have the same variable and same exponent, you can just look at the coefficients to tell you how many you have.

There’s no change to the variable part. You’re just changing the count of how many of that variable you have. The variable portions tell you that you’re working with the same kinds of things, and the coefficients tell you how many of them you have. If you were asked to add 7 cars and 5 cars, you’d get 12 cars, not 12 cars squared. 8 apples minus 3 apples gives you 5 apples. The cars or apples don’t change. When you add or subtract like terms, only the number changes.

WORLDLY WISDOM

When you add fractions, you need to have like denominators. When you add terms, you need to have like terms. The denominator of a fraction tells you what kind of fraction you have, and the numerator tells you how many of them you have. If you have the same denominators, you can just add the numerators, but if the denominators are different, you can't combine the fractions. The variable part of a term tells you what kind of thing you have, and the coefficient tells you how many. If the variable parts are the same, you can add the coefficients.

If you have to subtract terms, you follow the same rule: you can only subtract like terms, and you subtract the coefficients and keep the variable part exactly as it was. To subtract 8t - 11t, you subtract 8 - 11 to get -3, and you keep the t. So 8t - 11t = -3t.

If you’re faced with an addition or subtraction problem and you realize that the terms are unlike, you just leave the problem as it is, or you might say “this cannot be combined.”

CHECK POINT

Is it possible to complete these additions and subtractions? Complete them if you can!

11. -4x + 9x

12. 3a^{2} - 2a^{3}

13. 5xy + 6xy

14. 120xy^{2} — 80xy^{2}

15. 15z + 25x