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 \(812(6+4)\).
 \(8612+4\)
 \(81264\)
 \(812+(6+4)\)
 \(8126+4\)
 \(84126\)
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
 \((9x+5y) + (3x+7y)\)
 \((9x+5y)  (3x+7y)\)
 \((9x+5y)  (3x7y)\)
 \(9x7y + 3x+ 5y\)
 \(9x7y + 3x 5y\)
 \(9x7y  3x5y\)
B
 \(12(x+y)\)
 \(12(xy)\)
 \(6(x2y)\)
 \(9x+5y+3x7y\)
 \(9x+5y3x+7y\)
 \(9x3x+5y7y\)
22.3: Seeing Structure and Factoring
Write each expression with fewer terms. Show or explain your reasoning.

\(3 \boldcdot 15 + 4 \boldcdot 15  5 \boldcdot 15 \)

\(3x + 4x  5x\)

\(3(x2) + 4(x2)  5(x2) \)
 \(3\left(\frac52x+6\frac12\right) + 4\left(\frac52x+6\frac12\right)  5\left(\frac52x+6\frac12\right)\)
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:
\(\begin{gather} 2a3b4a5b \\ 2a+\text3b+\text4a+\text5b\\ 2a + \text4a + \text3b + \text5b\\ \text2a+\text8b\\ \text2a8b \\ \end{gather}\)

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:
\(\begin{gather} 5(x+3)0.5(x+3)+2(x+3)\\ (50.5+2)(x+3)\\ 6.5(x+3)\\ \end{gather}\)
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.