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.