# 2.2 Evaluate, simplify, and translate expressions  (Page 5/9)

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Geoffrey has dimes and quarters in his pocket. The number of dimes is seven less than six times the number of quarters. Let $q$ represent the number of quarters. Write an expression for the number of dimes.

6 q − 7

Lauren has dimes and nickels in her purse. The number of dimes is eight more than four times the number of nickels. Let $n$ represent the number of nickels. Write an expression for the number of dimes.

4 n + 8

## Key concepts

• Combine like terms.
1. Identify like terms.
2. Rearrange the expression so like terms are together.
3. Add the coefficients of the like terms

## Practice makes perfect

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.

$7x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=2$

22

$9x+7\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3$

$5x-4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6$

26

$8x-6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=7$

${x}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=12$

144

${x}^{3}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=5$

${x}^{5}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=2$

32

${x}^{4}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3$

${3}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3$

27

${4}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=2$

${x}^{2}+3x-7\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4$

21

${x}^{2}+5x-8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6$

$2x+4y-5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=7,y=8$

41

$6x+3y-9\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6,y=9$

${\left(x-y\right)}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=10,y=7$

9

${\left(x+y\right)}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6,y=9$

225

${a}^{2}+{b}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}a=3,b=8$

73

${r}^{2}-{s}^{2}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}r=12,s=5$

$2l+2w\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}l=15,w=12$

54

$2l+2w\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}l=18,w=14$

Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.

$15{x}^{2}+6x+2$

15 x 2 , 6 x , 2

$11{x}^{2}+8x+5$

$10{y}^{3}+y+2$

10 y 3 , y , 2

$9{y}^{3}+y+5$

In the following exercises, identify the coefficient of the given term.

$8a$

8

$13m$

$5{r}^{2}$

5

$6{x}^{3}$

In the following exercises, identify all sets of like terms.

${x}^{3},8x,14,8y,5,8{x}^{3}$

x 3 , 8 x 3 and 14, 5

$6z,3{w}^{2},1,6{z}^{2},4z,{w}^{2}$

$9a,{a}^{2},16ab,16{b}^{2},4ab,9{b}^{2}$

16 ab and 4 ab ; 16 b 2 and 9 b 2

$3,25{r}^{2},10s,10r,4{r}^{2},3s$

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.

$10x+3x$

13 x

$15x+4x$

$17a+9a$

26 a

$18z+9z$

$4c+2c+c$

7 c

$6y+4y+y$

$9x+3x+8$

12 x + 8

$8a+5a+9$

$7u+2+3u+1$

10 u + 3

$8d+6+2d+5$

$7p+6+5p+4$

12 p + 10

$8x+7+4x-5$

$10a+7+5a-2+7a-4$

22 a + 1

$7c+4+6c-3+9c-1$

$3{x}^{2}+12x+11+14{x}^{2}+8x+5$

17 x 2 + 20 x + 16

$5{b}^{2}+9b+10+2{b}^{2}+3b-4$

Translate English Phrases into Algebraic Expressions

In the following exercises, translate the given word phrase into an algebraic expression.

The sum of 8 and 12

8 + 12

The sum of 9 and 1

The difference of 14 and 9

14 − 9

8 less than 19

The product of 9 and 7

9 ⋅ 7

The product of 8 and 7

The quotient of 36 and 9

36 ÷ 9

The quotient of 42 and 7

The difference of $x$ and $4$

x − 4

$3$ less than $x$

The product of $6$ and $y$

6 y

The product of $9$ and $y$

The sum of $8x$ and $3x$

8 x + 3 x

The sum of $13x$ and $3x$

The quotient of $y$ and $3$

$\frac{y}{3}$

The quotient of $y$ and $8$

Eight times the difference of $y$ and nine

8 ( y − 9)

Seven times the difference of $y$ and one

Five times the sum of $x$ and $y$

5 ( x + y )

Nine times five less than twice $x$

In the following exercises, write an algebraic expression.

Adele bought a skirt and a blouse. The skirt cost $\text{15}$ more than the blouse. Let $b$ represent the cost of the blouse. Write an expression for the cost of the skirt.

b + 15

Eric has rock and classical CDs in his car. The number of rock CDs is $3$ more than the number of classical CDs. Let $c$ represent the number of classical CDs. Write an expression for the number of rock CDs.

The number of girls in a second-grade class is $4$ less than the number of boys. Let $b$ represent the number of boys. Write an expression for the number of girls.

b − 4

Marcella has $6$ fewer male cousins than female cousins. Let $f$ represent the number of female cousins. Write an expression for the number of boy cousins.

Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let $n$ represent the number of nickels. Write an expression for the number of pennies.

2 n − 7

Jeannette has $\text{5}$ and $\text{10}$ bills in her wallet. The number of fives is three more than six times the number of tens. Let $t$ represent the number of tens. Write an expression for the number of fives.

## Everyday math

In the following exercises, use algebraic expressions to solve the problem.

Car insurance Justin’s car insurance has a $\text{750}$ deductible per incident. This means that he pays $\text{750}$ and his insurance company will pay all costs beyond $\text{750.}$ If Justin files a claim for $\text{2,100,}$ how much will he pay, and how much will his insurance company pay?

He will pay $750. His insurance company will pay$1350.

Home insurance Pam and Armando’s home insurance has a $\text{2,500}$ deductible per incident. This means that they pay $\text{2,500}$ and their insurance company will pay all costs beyond $\text{2,500.}$ If Pam and Armando file a claim for $\text{19,400,}$ how much will they pay, and how much will their insurance company pay?

## Writing exercises

Explain why “the sum of x and y ” is the same as “the sum of y and x ,” but “the difference of x and y ” is not the same as “the difference of y and x .” Try substituting two random numbers for $x$ and $y$ to help you explain.

Explain the difference between $\text{“4}$ times the sum of $x$ and $y\text{”}$ and “the sum of $4$ times $x$ and $y\text{.”}$

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After reviewing this checklist, what will you do to become confident for all objectives?

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