Evaluate Algebraic 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.

Identify Terms, Coefficients, & Like Terms

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 [latex]x=1\cdot x[/latex]. 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
[latex]x+7[/latex]	x, 7
[latex]2x+7y+4[/latex]	2x, 7y, 4
[latex]3x^2+4x^2+5y+3[/latex]	3x2, 4x2 , 5y, 3

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.

Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think [latex]3x+6x[/latex] would simplify to? If you thought 9x, you would be right!

We can see why this works by writing both terms as addition problems.

[latex]\begin{array}{ccc}3x&+&6x \\ x+x+x &+& x+x+x+x+x+x \\ &9x& \end{array}[/latex]

Add the coefficients and keep the same variable. It doesn’t matter what x is. If you have 3 of something and add 6 more of the same thing, the result is 9 of them. For example, 3 oranges plus 6 oranges is 9 oranges. We will discuss the mathematical properties behind this later.

The expression [latex]3x+6x[/latex] has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together.

The Commutative Property of Addition says that we can change the order of addends without changing the sum. So, we could rearrange the following expression before combining like terms.

[latex]\begin{array}{ccc}\textcolor{red}{3x}+\textcolor{blue}{4y}&-&\textcolor{red}{2x}+\textcolor{blue}{6y} \\ &\Downarrow& \\ \textcolor{red}{3x}-\textcolor{red}{2x}&+&\textcolor{blue}{4y}+\textcolor{blue}{6y} \end{array}[/latex]

 

Now, it is easier to see the like terms to be combined.

Translate Words to Algebraic Expressions

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	[latex]a+b[/latex]
Subtraction	a minus b The difference of a and b b subtracted from a a decreased by b b less than a	[latex]a-b[/latex]
Multiplication	a times b The product of a and b	[latex]a\cdot b[/latex], ab, [latex]a(b)[/latex], [latex](a)(b)[/latex]
Division	a divided by b The quotient of a and b The ratio of a and b b divided into a	[latex]a \div b[/latex], [latex]a/b[/latex], [latex]\frac{a}{b}[/latex], [latex]b\overline{)a}[/latex]

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.

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.

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.

Key Concepts

Combine like terms

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

Glossary

• Term — A term is a constant or the product of a constant and one or more variables.
• Coefficient — The constant that multiplies the variable(s) in a term is called the coefficient.
• Like terms — Terms that are either constants or have the same variables with the same exponents are like terms.
• Evaluate — To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number.

8.2: Practice Questions

1. Evaluate Algebraic Expressions

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

1. [latex]7x+8[/latex] when [latex]x=2[/latex]
2. [latex]9x+7[/latex] when [latex]x=3[/latex]
3. [latex]5x-4[/latex] when [latex]x=6[/latex]
4. [latex]8x-6[/latex] when [latex]x=7[/latex]
5. [latex]x^2[/latex] when [latex]x=12[/latex]
6. [latex]6x^3[/latex] when [latex]x=5[/latex]
7. [latex]x^5[/latex]when [latex]x=2[/latex]
8. [latex]x^4[/latex] when [latex]x=3[/latex]
9. [latex]3^x[/latex] when [latex]x=3[/latex]
10. [latex]4^x[/latex]when [latex]x=2[/latex]
11. [latex]x^2+3x-7[/latex] when [latex]x=4[/latex]
12. [latex]x^2+5x-8[/latex] when [latex]x=6[/latex]
13. [latex]2x+4y-5[/latex] when [latex]x=7, y=8[/latex]
14. [latex]6x+3y-9[/latex] when [latex]x=6, y=9[/latex]
15. [latex]x-y^2[/latex] when [latex]x=10, y=7[/latex]
16. [latex]x+y^2[/latex] when [latex]x=6, y=9[/latex]
17. [latex]a^2+b^2[/latex] when [latex]a=3, b=8[/latex]
18. [latex]r^2-s^2[/latex] when [latex]r=12, s=5[/latex]
19. [latex]2l+2w[/latex] when [latex]l=15, w=12[/latex]
20. [latex]2l+2w[/latex] when [latex]l=18, w=14[/latex]

2. Identify Terms, Coefficients, and Like Terms

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

1. [latex]15x^2+6x+2[/latex]
2. [latex]11x^2+8x+5[/latex]
3. [latex]10y^3+y+2[/latex]
4. [latex]9y^3+y+5[/latex]

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

5. 8a
6. 13m
7. 5r²
8. 6x³

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

9. x³, 8x, 14, 8y, 5, 8x³
10. 6z, 3w², 1, 6z², 4z, w²
11. 9a, a², 16ab, 16b², 4ab, 9b²
12. 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.

1. [latex]10x+3x[/latex]
2. [latex]15x+4x[/latex]
3. [latex]17a+9a[/latex]
4. [latex]18z+9z[/latex]
5. [latex]4c+2c+c[/latex]
6. [latex]6y+4y+y[/latex]
7. [latex]9x+3x+8[/latex]
8. [latex]8a+5a+9[/latex]
9. [latex]7u+2+3u+1[/latex]
10. [latex]8d+6+2d+5[/latex]
11. [latex]7p+6+5p+4[/latex]
12. [latex]8x+7+4x-5[/latex]
13. [latex]10a+7+5a-2+7a-4[/latex]
14. [latex]7c+4+6c-3+9c-1[/latex]
15. [latex]3x^2+12x+11+14x^2+8x+5[/latex]
16. [latex]5b^2+9b+10+2b^2+3b-4[/latex]

4. Translate English Phrases into Algebraic Expressions

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

1. The sum of 8 and 12
2. The sum of 9 and 1
3. The difference of 14 and 9
4. 8 less than 19
5. The product of 9 and 7
6. The product of 8 and 7
7. The quotient of 36 and 9
8. The quotient of 42 and 7
9. The difference of x and 4
10. 3 less than x
11. The product of 6 and y
12. The product of 9 and y
13. The sum of 8x and 3x
14. The sum of 13x and 3x
15. The quotient of y and 3
16. The quotient of y and 8
17. Eight times the difference of y and nine
18. Seven times the difference of y and one
19. Five times the sum of x and y
20. Times five less than twice x

In the following exercises, write an algebraic expression.

21. 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.
22. 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.
23. 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.
24. 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.
25. 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.
26. 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.

1. 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?
2. 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

1. 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.
2. 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

1. Evaluate algebraic expressions.
   1. 22
   3. 26
   5. 144
   7. 32
   9. 27
   11. 21
   13. 41
   15. 9
   17. 73
   19. 54

    

2. 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

    

3. 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

    

4. 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. [latex]\dfrac{y}{3}[/latex]
   17. 8 (y − 9)
   19. 5 (x + y)
   21. b + 15
   23. b − 4
   25. 2n − 7

    

5. Everyday math.
   1. He will pay $750. His insurance company will pay $1350.

    

6. Writing exercises.
   1. Answers will vary.

References

Mathispower4u. (2014, August 10). Algebraic expression vocabulary (L2.2) [Video]. YouTube. https://youtu.be/E6lK-by__rA?si=YMB1kgHMXUfRy4ZI.

Attributions

This chapter has been adapted from 1.3 Evaluate, Simplify, and Translate Expressions in Introductory Algebra (BCcampus) by Izabela Mazur (2021), which is under a CC BY 4.0 license.

The original chapter was adapted from 2.2 Evaluate, Simplify, and Translate Expressions in Prealgebra 2e (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis (2020), which is under a CC BY 4.0 license. Adapted by Izabela Mazur.