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:

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