# Lesson 22

Combining Like Terms (Part 3)

Let’s see how we can combine terms in an expression to write it with less terms.

### 22.1: Are They Equal?

Select all expressions that are equal to $$8-12-(6+4)$$.

1. $$8-6-12+4$$
2. $$8-12-6-4$$
3. $$8-12+(6+4)$$
4. $$8-12-6+4$$
5. $$8-4-12-6$$

### 22.2: X’s and Y’s

Match each expression in column A with an equivalent expression from column B. Be prepared to explain your reasoning.

A

1. $$(9x+5y) + (3x+7y)$$
2. $$(9x+5y) - (3x+7y)$$
3. $$(9x+5y) - (3x-7y)$$
4. $$9x-7y + 3x+ 5y$$
5. $$9x-7y + 3x- 5y$$
6. $$9x-7y - 3x-5y$$

B

1. $$12(x+y)$$
2. $$12(x-y)$$
3. $$6(x-2y)$$
4. $$9x+5y+3x-7y$$
5. $$9x+5y-3x+7y$$
6. $$9x-3x+5y-7y$$

### 22.3: Seeing Structure and Factoring

Write each expression with fewer terms. Show or explain your reasoning.

1. $$3 \boldcdot 15 + 4 \boldcdot 15 - 5 \boldcdot 15$$

2. $$3x + 4x - 5x$$

3. $$3(x-2) + 4(x-2) - 5(x-2)$$

4. $$3\left(\frac52x+6\frac12\right) + 4\left(\frac52x+6\frac12\right) - 5\left(\frac52x+6\frac12\right)$$

### Summary

Combining like terms is a useful strategy that we will see again and again in our future work with mathematical expressions. It is helpful to review the things we have learned about this important concept.

• Combining like terms is an application of the distributive property. For example:

$$\begin{gather} 2x+9x\\ (2+9) \boldcdot x \\ 11x\\ \end{gather}$$

• It often also involves the commutative and associative properties to change the order or grouping of addition. For example:

$$\begin{gather} 2a+3b+4a+5b \\ 2a+4a+3b+5b \\ (2a+4a)+(3b+5b) \\ 6a+8b\\ \end{gather}$$

• We can't change order or grouping when subtracting; so in order to apply the commutative or associative properties to expressions with subtraction, we need to rewrite subtraction as addition. For example:

$$\begin{gather} 2a-3b-4a-5b \\ 2a+\text-3b+\text-4a+\text-5b\\ 2a + \text-4a + \text-3b + \text-5b\\ \text-2a+\text-8b\\ \text-2a-8b \\ \end{gather}$$

• Since combining like terms uses properties of operations, it results in expressions that are equivalent.

• The like terms that are combined do not have to be a single number or variable; they may be longer expressions as well. Terms can be combined in any sum where there is a common factor in all the terms. For example, each term in the expression $$5(x+3)-0.5(x+3)+2(x+3)$$ has a factor of $$(x+3)$$. We can rewrite the expression with fewer terms by using the distributive property:

$$\begin{gather} 5(x+3)-0.5(x+3)+2(x+3)\\ (5-0.5+2)(x+3)\\ 6.5(x+3)\\ \end{gather}$$

### Glossary Entries

• expand

To expand an expression, we use the distributive property to rewrite a product as a sum. The new expression is equivalent to the original expression.

For example, we can expand the expression $$5(4x+7)$$ to get the equivalent expression $$20x + 35$$.

• factor (an expression)

To factor an expression, we use the distributive property to rewrite a sum as a product. The new expression is equivalent to the original expression.

For example, we can factor the expression $$20x + 35$$ to get the equivalent expression $$5(4x+7)$$.

• term

A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression $$5x + 18$$ has two terms. The first term is $$5x$$ and the second term is 18.