Introduction

To simplify an expression containing parentheses, we expand the parentheses and then collect like terms.

For example, let's simplify the following expression:

3(x + 2) + 4x

Our first step is to distribute the 3 over each of the terms in the parentheses:

\eqalign{ 3(x + 2) + 4x &= \\[5pt] 3\cdot x + 3\cdot 2 + 4x &= \\[5pt] (3x) + (6) + 4x &= \\[5pt] 3x +6 + 4x & }

The second and final step is to collect the like terms:

\eqalign{ 3x + 6 + 4x &= \\ (3x + 4x) + 6 &= \\ 7x + 6 }

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Example: Simplifying Linear Expressions With Positive Numbers in Parentheses

Simplify (4+c)\cdot 3-8.

EXPLANATION

First, we distribute the 3 over each of the terms in parentheses.

\eqalign{ (4+c)\cdot 3-8 &= \\[5pt] 4\cdot3+c\cdot 3-8 &= \\[5pt] (12)+(3c)-8 &= \\[5pt] 12+3c-8 & }

Then, we collect like terms.

\eqalign{ 12+3c-8 &= \\[5pt] 3c +(12-8)&= \\[5pt] 3c+4 }

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Practice: Simplifying Linear Expressions With Positive Numbers in Parentheses

$-4x + 3(7x+y) =$

a
 $3x+y$
b
 $3y-17x$
c
 $17x+3y$
d
 $17x+y$
e
 $21x+3y$
Practice: Simplifying Linear Expressions With Positive Numbers in Parentheses

$3u +(2u+v)\cdot4 =$

a
 $8u+4v$
b
 $11u+v$
c
 $2u+v$
d
 $3u-4v$
e
 $11u+4v$
Example: Simplifying Linear Expressions With Positive Numbers in Parentheses: Distributing a Negative Number

(-4)(8x+7y) +30y=

EXPLANATION

First, we distribute the (-4) over each of the terms in parentheses.

\begin{eqnarray} \eqalign{ (-4)(8x+7y) +30y&=\\[5pt] (-4)\cdot 8x + (-4)\cdot 7y +30y &=\\[5pt] (-32x) + (-28y) +30y &=\\[5pt] -32x -28y+30y } \end{eqnarray}

Then, we collect like terms.

\begin{eqnarray} \eqalign{ -32x -28y+30y&=\\[5pt] -32x+( -28y+30y)&=\\[5pt] -32x +2y } \end{eqnarray}

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Practice: Simplifying Linear Expressions With Positive Numbers in Parentheses: Distributing a Negative Number

$(-3)(6x+y) -4x=$

a
 $22x + 3y$
b
 $-22x - 3y$
c
 $22x + 6y$
d
 $-22x + 6y$
e
 $14x - 3y$
Practice: Simplifying Linear Expressions With Positive Numbers in Parentheses: Distributing a Negative Number

$2u +(5u+v)\cdot(-4) =$

a
 $16u - 4v$
b
 $-16u + 4v$
c
 $-18u - 4v$
d
 $-16u - 4v$
e
 $18u - 4v$
Example: Simplifying Linear Expressions With Negative Numbers in Parentheses

Simplify (5 - y) \cdot 3 - 10

EXPLANATION

First, we distribute the 3 over each of the terms in parentheses.

\begin{align} (5 - y) \cdot 3 - 10& = \\[5pt] 5\cdot 3 + (-y) \cdot 3 - 10 & = \\[5pt] (15) + (-3y) - 10& = \\[5pt] 15 - 3y - 10 \end{align}

Then, we collect like terms.

\begin{align} 15 - 3y - 10 & = \\[5pt] (15 - 10) - 3y& = \\[5pt] 5 - 3y \end{align}

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Practice: Simplifying Linear Expressions With Negative Numbers in Parentheses

$3(x-5)-x=$

a
 $2x+5$
b
 $4x-5$
c
 $2x+15$
d
 $2x-5$
e
 $2x-15$
Practice: Simplifying Linear Expressions With Negative Numbers in Parentheses

$6g+(f-2g)\cdot2=$

a
 $f+2g$
b
 $2f+2g$
c
 $2f+10g$
d
 $f-2g$
e
 $4fg$
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