# Expressions and Equations

### Simplifying Expressions

#### Terms

A term is the product of a number and one or more variables. A term may also be a single number, with no variable.
The expression 14 + 3y 2 - 15zp has three terms: 14 , 3y 2 , and -15zp .
The expression 7z + 12 + 2 + z has four terms: 7z , 12, 2, z .

The coefficient is the number that is multiplied by the variable(s) in a single term. In the expression 14 + 3y 2 - 15zp , y 2 has a coefficient of 3 and zp has a coefficient of -15 . A term with no coefficient, like z , has an implied coef ficient of 1 .

Terms that do not contain variables are called constants. In the above expressions, 14 , 12 , and 2 are constants.

#### Combining Like Terms

Like terms are terms that contain the exact same variables raised to the same exponents. For example, 15yz and 22yz are like terms, but 15yz 2 and 22yz are not. Likewise, 12w 2 yz and -5w 2 yz are like terms, but 12w 2 yz and -5w 2 z are not. 12 and -6 are like terms, because they are both constant terms.

To combine like terms, group them together in the equation, putting the terms with the highest exponents on the left. We can do this because addition commutes. Next, group the coefficients of like terms together , all multiplied by the variable(s) in those terms. Finally, add the coefficients of the like terms (or subtract them if they are negative). Here are some examples:

Example 1: Simplify 4y + 15 - 2y + 5y 2 + 12 - 6 .

1. 5y 2 + 4y - 2y + 15 + 12 - 6 (note that this expression and the original expression both contain 6 terms)
2. 5y 2 + (4 - 2)y + (15 + 12 - 6)
3. 5y 2 + 2y + 21

Example 2: Simplify 11wz + 6w 2 -10wz + wz - 6 - 2w + 3w 2 .
1. 6w 2 +3w 2 + 11wz - 10wz + wz - 2w - 6 (note that this expression and the original expression both contain 7 terms)
2. (6 + 3)w 2 + (11 - 10 + 1)wz - 2w - 6
3. 9w 2 + 2wz - 2w - 6

Example 3: Simplify -12w 2 +6w - 7w 3 +4w 2 +8w 2 - 2w + 5 + 9w .
1. -7w 3 -12w 2 +4w 2 +8w 2 + 6w - 2w + 9w + 5 (8 terms)
2. -7w 3 + (- 12 + 4 + 8)w 2 + (6 - 2 + 9)w + 5
3. -7w 3 +0w 2 +13w + 5 = - 7w 3 + 13w + 5

When we combine like terms, we convert the expression to simplified form. The expression is simplified form is equivalent to the original expression.

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