8.2: Evaluate, Simplify, & Translate Expressions

Kim Tamblyn

8.2: Evaluate, Simplify, & Translate Expressions

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions — again following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

Example A

Evaluate $x+7$ when:

$x=3$
$x=12$

a. $\boldsymbol{x=3}$

To evaluate, substitute 3 for x in the expression, and then simplify.

$\begin{array}{rr} &x+7 \\ \text{Substitute:}&3+7 \\ \text{Add:}&10 \end{array}$

When $x=3$ , the expression $x+7$ has a value of 10.

b. $\boldsymbol{x=12}$

To evaluate, substitute 12 for x in the expression, and then simplify.

$\begin{array}{rr} &x+7 \\ \text{Substitute:}&12+7 \\ \text{Add:}&19 \end{array}$

When $x=12$ , the expression $x+7$ has a value of 19.

Notice that we got different results for parts a) and b) even though we started with the same expression. This is because the values used for x were different. When we evaluate an expression, the value varies depending on the value used for the variable.

Exercise 1

Evaluate $y+4$ when:

$y=6$
$y=15$

Exercise 1 Answers

10
19

Exercise 2

Evaluate $a-5$ when:

$a=9$
$a=17$

Exercise 2 Answers

4
12

Example B

Evaluate $9x-2$ when:

$x=5$
$x=1$

Remember ab means a times b, so 9x means 9 times x.

a. $\boldsymbol{x=5}$

To evaluate the expression when $x=5$ , we substitute 5 for x, and then simplify.

$\begin{array}{rr}&9x-2 \\ \text{Substitute } 5 \text{ for } x : &9\cdot 5 - 2 \\ \text{Multiply:} &45-2 \\ \text{Subtract:} &43 \end{array}$

b. $\boldsymbol{x=1}$

To evaluate the expression when $x=1$ , we substitute 1 for x, and then simplify.

$\begin{array}{rr}&9x-2 \\ \text{Substitute } 1 \text{ for } x : &9(1) - 2 \\ \text{Multiply:} &9-2 \\ \text{Subtract:} &7 \end{array}$

Notice that in part a), we wrote $9\cdot 5$ , and in part b), we wrote $9(1)$ . Both the dot and the parentheses tell us to multiply.

Exercise 3

Evaluate $8x-3$ when:

$x=2$
$x=1$

Exercise 3 Answers

13
5

Exercise 4

Evaluate $4y-4$ when:

$y=3$
$y=5$

Exercise 4 Answers

8
16

Example C

Evaluate $x^2$ when $x=10$ .

We substitute 10 for x, and then simplify the expression.

$\begin{array}{rr}&x^2 \\ \text{Substitute } 10 \text{ for } x : &10^2 \\ \text{Use the definition of exponent:} &10\cdot 10 \\ \text{Multiply:} &100 \end{array}$

When $x=10$ , the expression $x^2$ has a value of 100.

Exercise 5

Evaluate $x^2$ when $x=8$ .

Exercise 5 Answer

64

Exercise 6

Evaluate $x^3$ when $x=6$ .

Exercise 6 Answer

216

Example D

Evaluate $2^x$ when $x=5$ .

In this expression, the variable is an exponent.

$\begin{array}{rr}&2^x \\ \text{Substitute } 5 \text{ for } x : &2^5 \\ \text{Use the definition of exponent:} &2\cdot 2\cdot 2\cdot 2\cdot 2 \\ \text{Multiply:} &32 \end{array}$

When $x=5$ , the expression $2^x$ has a value of 32.

Exercise 7

Evaluate $2^x$ when $x=6$ .

Exercise 7 Answer

64

Exercise 8

Evaluate $3^x$ when $x=4$ .

Exercise 8 Answer

81

Example E

Evaluate $3x+4y-6$ when $x=10$ and $y=2$ .

This expression contains two variables, so we must make two substitutions.

$\begin{array}{rr}&3x+4y-6 \\ \text{Substitute } 10 \text{ for } x \text{ and } 2 \text{ for } y : &3(10)+4(2)-6 \\ \text{Multiply:} &30+8-6 \\ \text{Add and subtract left to right:} &32 \end{array}$

When $x=10$ and $y=2$ , the expression $3x+4y-6$ has a value of 32.

Exercise 9

Evaluate $2x+5y-4$ when $x=11$ and $y=3$ .

Exercise 9 Answer

33

Exercise 10

Evaluate $5x-2y-9$ when $x=7$ and $y=8$ .

Exercise 10 Answer

10

Example F

Evaluate $2x^2+3x+8$ when $x=4$ .

We need to be careful when an expression has a variable with an exponent. In this expression, $2x^2$ means $2\cdot x\cdot x$ and is different from the expression $(2x)^2$ , which means $2x\cdot 2x$ .

$\begin{array}{rr}&2x^2+3x+8 \\ \text{Substitute } 4 \text{ for } x : &2(4)^2+3(4)+8 \\ \text{Simplify }4^2 : &2(16)+3(4)+8 \\ \text{Multiply:} &32+12+8 \\ \text{Add: } &52 \end{array}$

Exercise 11

Evaluate $3x^2+4x+1$ when $x=3$ .

Exercise 11 Answer

40

Exercise 12

Evaluate $6x^2-4x-7$ when $x=2$ .

Exercise 12 Answer

9

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are 7, y, 5x², 9a, and 13xy.

The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term 3x is 3. When we write x, the coefficient is 1 since $x=1\cdot x$ . Table 8.2.1 gives the coefficients for each of the terms in the left column.

Table 8.2.1: Identifying Coefficients of Terms
Term	Coefficient
7	7
9a	9
y	1
5x²	5

An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. Table 8.2.2 gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

Table 8.2.2: Identifying Terms in an Expression
Expression	Terms
7	7
y	y
$x+7$	x, 7
$2x+7y+4$	2x, 7y, 4
$3x^2+4x^2+5y+3$	3x², 4x² , 5y, 3

Example G

Identify each term in the expression $9b+15x^2+a+6$ . Then, identify the coefficient of each term.

The expression has four terms. They are 9b, 15x², a, and 6.

The coefficient of 9b is 9.
The coefficient of 15x² is 15.
Remember that if no number is written before a variable, the coefficient is 1. So, the coefficient of a is 1.
The coefficient of a constant is the constant, so the coefficient of 6 is 6.

Exercise 13

Identify all terms in the given expression, and their coefficients: $4x+3b+2$ .

Exercise 13 Answer

The terms are 4x, 3b, and 2.
The coefficients are 4, 3, and 2.

Exercise 14

Identify all terms in the given expression, and their coefficients: $9a+13a^2+a^3$ .

Exercise 14 Answer

The terms are 9a, 13a², and a³.
The coefficients are 9, 13, and 1.

Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

5x, 7, n², 4, 3x, 9n²

Which of these terms are like terms?

The terms 7 and 4 are both constant terms.
The terms 5x and 3x are both terms with x.
The terms n² and 9n² both have n².

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms 5x, 7, n², 4, 3x, 9n²:

7 and 4 are like terms.
5x and 3x are like terms.
n² and 9n² are like terms.

Like terms:

Terms that are either constants or have the same variables with the same exponents are like terms.

Example H

Identify the like terms:

$y^3, 7x^2, 14, 23, 4y^3, 9x, 5x^2$
$4x^2+2x+5x^2+6x+40x+8xy$

a. $\boldsymbol{y^3, 7x^2, 14, 23, 4y^3, 9x, 5x^2}$

Look at the variables and exponents. The expression contains y³, x², x, and constants.

The terms y³ and 4y³ are like terms because they both have y³.
The terms 7x² and 5x² are like terms because they both have x².
The terms 14 and 23 are like terms because they are both constants.
The term 9x does not have any like terms in this list since no other terms have the variable x raised to the power of 1.

b. $\boldsymbol{4x^2+2x+5x^2+6x+40x+8xy}$

Look at the variables and exponents. The expression contains the terms 4x², 2x, 5x², 6x, 40x, and 8xy.

The terms 4x² and 5x² are like terms because they both have x².
The terms 2x, 6x, and 40x are like terms because they all have x.
The term 8xy has no like terms in the given expression because no other terms contain the two variables xy.

Exercise 15

Identify the like terms in the list or the expression: $9, 2x^3, y^2, 8x^3, 15, 9y, 11y^2$

Exercise 15 Answers

9 and 15; 2x³ and 8x³; y², and 11y²

Exercise 16

Identify the like terms in the list or the expression: $4x^3+8x^2+19+3x^2+24+6x^3$

Exercise 16 Answers

4x³ and 6x³; 8x² and 3x²; 19 and 24

In the previous section, we listed many operation symbols used in algebra, and then we translated expressions and equations into word phrases and sentences. Now, we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. Table 8.2.3 summarizes them.

Table 8.2.3: Translating Words to Algebraic Expressions
Operation	Phrase	Expression
Addition	a plus b The sum of a and b a increased by b b more than a The total of a and b b added to a	$a+b$
Subtraction	a minus b The difference of a and b b subtracted from a a decreased by b b less than a	$a-b$
Multiplication	a times b The product of a and b	$a\cdot b$ , ab, $a(b)$ , $(a)(b)$
Division	a divided by b The quotient of a and b The ratio of a and b b divided into a	$a \div b$ , $a/b$ , $\frac{a}{b}$ , $b\overline{)a}$

Look closely at these phrases using the four operations:

The sum of a and b
The difference of a and b
The product of a and b
The quotient of a and b

Each phrase tells you to operate on two numbers. Look for the words of and and to find the numbers.

Example L

Translate each word phrase into an algebraic expression:

The difference of 20 and 4
The quotient of 10x and 3

a. The difference of 20 and 4

The key word is difference, which tells us the operation is subtraction. Look for the words of and and to find the numbers to subtract.

$\begin{array}{l} \text{the difference of } 20 \text{ and } 4 \\ 20 \text{ minus } 4 \\ 20-4 \end{array}$

b. The quotient of 10x and 3

The key word is quotient, which tells us the operation is division.

$\begin{array}{l}\\ \text{the quotient of } 10x \text{ and } 3 \\ \text{divide } 10x \text{ by } 3 \\ 10x\div 3 \end{array}$

This can also be written as $\begin{array}{l}10x/3 \text{ or } \dfrac{10x}{3} \end{array}$ .

Exercise 21

Translate the given word phrase into an algebraic expression.

The difference of 47 and 41
The quotient of 5x and 2

Exercise 21 Answers

$47 − 41$
$5x \div 2$

Exercise 22

Translate the given word phrase into an algebraic expression.

The sum of 17 and 19
The product of 7 and x

Exercise 22 Answers

$17 + 19$
$7x$

How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eight more than means eight added to your present age.

How old were you seven years ago? This is seven years less than your age now. You subtract 7 from your present age. Seven less than means seven subtracted from your present age.

Example M

Translate each word phrase into an algebraic expression:

Eight more than y
Seven less than 9z

a. Eight more than y

The keywords are more than. They tell us the operation is addition. More than means “added to”.

$\begin{array}{l}\text{Eight more than}\phantom{\rule{0.2em}{0ex}}y\\ \text{Eight added to}\phantom{\rule{0.2em}{0ex}}y\\ y+8\end{array}$

b. Seven less than 9z

The keywords are less than. They tell us the operation is subtraction. Less than means “subtracted from”.

$\begin{array}{l}\text{Seven less than}\phantom{\rule{0.2em}{0ex}}9z\\ \text{Seven subtracted from}\phantom{\rule{0.2em}{0ex}}9z\\ 9z-7\end{array}$

Exercise 23

Translate each word phrase into an algebraic expression.

Eleven more than x
Fourteen less than 11a

Exercise 23 Answers

$x + 11$
$11a − 14$

Exercise 24

Translate each word phrase into an algebraic expression.

19 more than j
21 less than 2x

Exercise 24 Answers

$j + 19$
$2x − 21$

Example N

Translate each word phrase into an algebraic expression:

Five times the sum of m and n
The sum of five times m and n

a. Five times the sum of m and n

There are two operation words: times tells us to multiply, and sum tells us to add. Because we are multiplying 5 times the sum, we need parentheses around the sum of m and n.

$m+n$

b. The sum of five times m and n

To take a sum, we look for the words of and and to see what is being added. Here, we are taking the sum of five times m and n.

$5m+n$

Notice how the use of parentheses changes the result. In part a), we add first and in part b), we multiply first.

Exercise 25

Translate the word phrase into an algebraic expression.

Four times the sum of p and q
The sum of four times p and q

Exercise 25 Answers

$4(p + q)$
$4p + q$

Exercise 26

Translate the word phrase into an algebraic expression.

The difference of two times x and 8
Two times the difference of x and 8

Exercise 26 Answers

$2x − 8$
$2(x − 8)$

Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase into an algebraic expression. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples.

Example O

The height of a rectangular window is 6 inches less than the width. Let w represent the width of the window. Write an expression for the height of the window.

Step 1: Write a phrase about the height.

6 less than the width

Step 2: Substitute w for the width.

6 less than w

Step 3: Rewrite ‘less than’ as ‘subtracted from.’

6 subtracted from w

Step 4: Translate the phrase into algebra.

$w-6$

Exercise 27

The length of a rectangle is 5 inches less than the width. Let w represent the width of the rectangle. Write an expression for the length of the rectangle.

Exercise 27 Answer

$w − 5$

Exercise 28

The width of a rectangle is 2 metres greater than the length. Let l represent the length of the rectangle. Write an expression for the width of the rectangle.

Exercise 27 Answer

$l + 2$

Example P

Blanca has dimes and quarters in her purse. The number of dimes is 2 less than 5 times the number of quarters. Let q represent the number of quarters. Write an expression for the number of dimes.

Step 1: Write a phrase about the number of dimes.

Two less than five times the number of quarters

Step 2: Substitute q for the number of quarters.

2 less than five times q

Step 3: Translate 5 times q.

2 less than 5q

Step 4: Translate the phrase into algebra.

$5q-2$

Exercise 29

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.

Exercise 29 Answer

$6q − 7$

Exercise 30

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.

Exercise 30 Answer

$4n + 8$

Additional online resources:

Algebraic Expression Vocabulary — Algebraic Expression Vocabulary (L2.2) [5:52 min] (Mathispower4u, 2014).

1. Evaluate Algebraic Expressions

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

$7x+8$ when $x=2$
$9x+7$ when $x=3$
$5x-4$ when $x=6$
$8x-6$ when $x=7$
$x^2$ when $x=12$
$6x^3$ when $x=5$
$x^5$ when $x=2$
$x^4$ when $x=3$
$3^x$ when $x=3$
$4^x$ when $x=2$
$x^2+3x-7$ when $x=4$
$x^2+5x-8$ when $x=6$
$2x+4y-5$ when $x=7, y=8$
$6x+3y-9$ when $x=6, y=9$
$x-y^2$ when $x=10, y=7$
$x+y^2$ when $x=6, y=9$
$a^2+b^2$ when $a=3, b=8$
$r^2-s^2$ when $r=12, s=5$
$2l+2w$ when $l=15, w=12$
$2l+2w$ when $l=18, w=14$

2. Identify Terms, Coefficients, and Like Terms

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

$15x^2+6x+2$
$11x^2+8x+5$
$10y^3+y+2$
$9y^3+y+5$

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

8a
13m
5r²
6x³

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

x³, 8x, 14, 8y, 5, 8x³
6z, 3w², 1, 6z², 4z, w²
9a, a², 16ab, 16b², 4ab, 9b²
3, 25r², 10s, 10r, 4r², 3s

3. Simplify Expressions by Combining Like Terms

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

$10x+3x$
$15x+4x$
$17a+9a$
$18z+9z$
$4c+2c+c$
$6y+4y+y$
$9x+3x+8$
$8a+5a+9$
$7u+2+3u+1$
$8d+6+2d+5$
$7p+6+5p+4$
$8x+7+4x-5$
$10a+7+5a-2+7a-4$
$7c+4+6c-3+9c-1$
$3x^2+12x+11+14x^2+8x+5$
$5b^2+9b+10+2b^2+3b-4$

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
The sum of 9 and 1
The difference of 14 and 9
8 less than 19
The product of 9 and 7
The product of 8 and 7
The quotient of 36 and 9
The quotient of 42 and 7
The difference of x and 4
3 less than x
The product of 6 and y
The product of 9 and y
The sum of 8x and 3x
The sum of 13x and 3x
The quotient of y and 3
The quotient of y and 8
Eight times the difference of y and nine
Seven times the difference of y and one
Five times the sum of x and y
Times five less than twice x

In the following exercises, write an algebraic expression.

Adele bought a skirt and a blouse. The skirt cost $15 more than the blouse. Let b represent the cost of the blouse. Write an expression for the cost of the skirt.
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.
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.
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.
Jeannette has $5 and $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.

5. Everyday Math

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

Car insurance: Justin’s car insurance has a $750 deductible per incident. This means that he pays $750, and his insurance company will pay all costs beyond $750. If Justin files a claim for $2,100, how much will he pay, and how much will his insurance company pay?
Home insurance: Pam and Armando’s home insurance has a $2,500 deductible per incident. This means that they pay $2,500, and their insurance company will pay all costs beyond $2,500. If Pam and Armando file a claim for $19,400, how much will they pay, and how much will their insurance company pay?

6. 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 “4 times the sum of x and y” and “the sum of 4 times x and y.”

8.2: Practice Answers

Evaluate algebraic expressions.
1. 22
3. 26
5. 144
7. 32
9. 27
11. 21
13. 41
15. 9
17. 73
19. 54
Identify terms, coefficients, and like terms.
1. 15x², 6x, 2
3. 10y³, y, 2
5. 8
7. 5
9. x³, 8x³ and 14, 5
11. 16ab and 4ab; 16b² and 9b²
Simplify expressions by combining like terms.
1. 13x
3. 26a
5. 7c
7. 12x + 8
9. 10u + 3
11. 12p + 10
13. 22a + 1
15. 17x² + 20x + 16
Translate English phrases into algebraic expressions.
1. 8 + 12
3. 14 − 9
5. 9 ⋅ 7
7. 36 ÷ 9
9. x − 4
11. 6y
13. 8x + 3x
15. $\dfrac{y}{3}$
17. 8 (y − 9)
19. 5 (x + y)
21. b + 15
23. b − 4
25. 2n − 7
Everyday math.
1. He will pay $750. His insurance company will pay $1350.
Writing exercises.
1. Answers will vary.

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Intermediate PreAlgebra: Building Success Copyright © 2024 by Kim Tamblyn, TRU Open Press is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Evaluate Algebraic Expressions

Exercise 1 Answers

Exercise 2 Answers

Exercise 3 Answers

Exercise 4 Answers

Exercise 5 Answer

Exercise 6 Answer

Exercise 7 Answer

Exercise 8 Answer

Exercise 9 Answer

Exercise 10 Answer

Exercise 11 Answer

Exercise 12 Answer

Identify Terms, Coefficients, & Like Terms

Exercise 13 Answer

Exercise 14 Answer

Exercise 15 Answers

Exercise 16 Answers

Simplify Expressions by Combining Like Terms

Exercise 17 Answer

Exercise 18 Answer

Exercise 19 Answer

Exercise 20 Answer

Translate Words to Algebraic Expressions

Exercise 21 Answers

Exercise 22 Answers

Exercise 23 Answers

Exercise 24 Answers

Exercise 25 Answers

Exercise 26 Answers

Exercise 27 Answer

Exercise 27 Answer

Exercise 29 Answer

Exercise 30 Answer

Key Concepts

Combine like terms

Glossary

8.2: Practice Questions

1. Evaluate Algebraic Expressions

2. Identify Terms, Coefficients, and Like Terms

3. Simplify Expressions by Combining Like Terms

4. Translate English Phrases into Algebraic Expressions

5. Everyday Math

6. Writing Exercises

8.2: Practice Answers

References

Attributions

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