Lesson 22

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. \(8-4-12-6\)

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

 

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

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: 

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