Introduction

Remember that two terms are called like terms if they have the same variable part.

An expression with different terms can often be simplified by collecting like terms.

For example, consider the following expression: 2x+3x

The terms 2x and 3x are like terms because they both have a variable part of x.

Whenever we have two or more like terms in an expression, we can combine them into a single term. Here, we can combine the like terms 2x and 3x into a single term by adding their coefficients: \begin{align*} {\color{blue}2}x + {\color{blue}3}x&=\\[5pt] ({\color{blue}2}+{\color{blue}3})x&= \\[5pt] {\color{blue}5}x& \end{align*}

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Example: Simplifying Expressions With Two or Three Terms

Simplify the expression 2 + 7p - 3p.

EXPLANATION

We have two like terms, 7p and -3p.

To collect these like terms, we combine them into a single term, adding their coefficients:

\begin{align} 2 + 7p - 3p &=\\[5pt] 2 + (7-3)p &=\\[5pt] 2 + 4p& \end{align}

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Practice: Simplifying Expressions With Two or Three Terms

$5x+6x= $

a
 $11x$
b
 $11 + x$
c
 $5+11x$
d
 $11+2x$
e
 $30x$
Practice: Simplifying Expressions With Two or Three Terms

$7 + 5x - 2x = $

a
 $x$
b
 $-3x$
c
 $7+3x$
d
 $7-3x$
e
 $3x$
Practice: Simplifying Expressions With Two or Three Terms

$3x - 4z + 3z =$

a
 $10xz$
b
 $6x-4z$
c
 $3x -z$
d
 $7x+3z$
e
 $-4z$
Example: Simplifying Expressions With Four Terms

Simplify the expression 5k - 3k + 2k - t.

EXPLANATION

We have three like terms, 5k, -3k, and 2k.

To collect these like terms, we combine them into a single term, adding their coefficients:

\begin{align} 5k - 3k + 2k - t&=\\[5pt] (5 - 3 + 2)k - t & =\\[5pt] 4k - t \end{align}

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Practice: Simplifying Expressions With Four Terms

$2a + 3a - a + 4a =$

a
 $9a$
b
 $10a$
c
 $4a$
d
 $0$
e
 $8a$
Practice: Simplifying Expressions With Four Terms

$x + 3x - 2x + 4=$

a
 $2x + 4$
b
 $-3x + 4$
c
 $-2x + 4$
d
 $3x + 4$
e
 $x + 4$
Example: Rearranging an Expression Before Collecting Like Terms

Simplify the expression 6a+1-2a.

EXPLANATION

We rearrange the terms so that the like terms, 6a and -2a , are together.

\eqalign{ 6a+1-2a &= \\[5pt] \underbrace{6a-2a}_{a \textrm{ terms}}+ 1 & }

Then, we combine the like terms:

\eqalign{ \underbrace{6a-2a}+ 1 &= \\[5pt] (6-2)a + 1 &=\\[5pt] 4a + 1& }

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Practice: Rearranging an Expression Before Collecting Like Terms

$4x + 5y + x=$

a
 $5x + 5y$
b
 $4x + 5y$
c
 $5x + 4y$
d
 $4x - 5y$
e
 $5x - 5y$
Practice: Rearranging an Expression Before Collecting Like Terms

$13x - 2y + 5x + 1=$

a
 $13x -2y - 1$
b
 $11x + 5y + 1$
c
 $8x - 2y + 1$
d
 $18x+2y-1$
e
 $18x-2y+1$
Example: Collecting Groups of Like Terms

Simplify the expression 2.5x+2y-x+1.5y.

EXPLANATION

First, we group the like terms. \eqalign{ 2.5x+2y-x+1.5y &= \\[5pt] \underbrace{2.5x-x}_{x \textrm{ terms}}+\underbrace{2y+1.5y}_{y \textrm{ terms}} & }

Then, we combine the like terms: \eqalign{ \underbrace{2.5x-x}+\underbrace{2y+1.5y} &= \\[5pt] (2.5-1)x + (2+1.5)y &=\\[5pt] 1.5x + 3.5y }

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Practice: Collecting Groups of Like Terms

$3y + 4z - y + z=$

a
 $7y$
b
 $-2y+5z$
c
 $2y + 5z$
d
 $-2y - 5z$
e
 $2y - 5z$
Practice: Collecting Groups of Like Terms

$3.1t + 12 + 2.5t - 5 =$

a
 $5.6t$
b
 $5t + 7$
c
 $5.6t + 7$
d
 $6.2t + 7$
e
 $6t + 7$
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