### Learning Objectives

By the end of this section, you will be able to:

- Evaluate algebraic expressions
- Identify terms, coefficients, and like terms
- Simplify expressions by combining like terms
- Translate word phrases to algebraic expressions

Before you get started, take this readiness quiz.

Is $n\xf75$ an expression or an equation?

If you missed this problem, review Example 2.4.

Simplify ${4}^{5}.$

If you missed this problem, review Example 2.7.

Simplify $1+8\cdot 9.$

If you missed this problem, review Example 2.8.

### 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.

### Example 2.13

Evaluate $x+7$ when

- ⓐ $\phantom{\rule{0.2em}{0ex}}x=3$
- ⓑ $\phantom{\rule{0.2em}{0ex}}x=12$

Evaluate:

$y+4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

- ⓐ $\phantom{\rule{0.2em}{0ex}}y=6\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
- ⓑ $\phantom{\rule{0.2em}{0ex}}y=15$

Evaluate:

$a-5\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}$

- ⓐ $\phantom{\rule{0.2em}{0ex}}a=9\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
- ⓑ $\phantom{\rule{0.2em}{0ex}}a=17$

### Example 2.14

Evaluate $9x-2,\text{when}\phantom{\rule{0.2em}{0ex}}$

- ⓐ$\phantom{\rule{0.2em}{0ex}}x=5\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
- ⓑ $\phantom{\rule{0.2em}{0ex}}x=1$

Evaluate:

$8x-3,\text{when}\phantom{\rule{0.2em}{0ex}}$

- ⓐ$\phantom{\rule{0.2em}{0ex}}x=2\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
- ⓑ $\phantom{\rule{0.2em}{0ex}}x=1$

Evaluate:

$4y-4,\text{when}\phantom{\rule{0.2em}{0ex}}$

- ⓐ $\phantom{\rule{0.2em}{0ex}}y=3\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}$
- ⓑ $\phantom{\rule{0.2em}{0ex}}y=5$

### Example 2.15

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

Evaluate:

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

Evaluate:

${x}^{3}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6.$

### Example 2.16

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

Evaluate:

${2}^{x}\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=6.$

Evaluate:

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

### Example 2.17

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}3x+4y-6\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=2.$

Evaluate:

$2x+5y-4\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=11\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=3$

Evaluate:

$5x-2y-9\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=7\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=8$

### Example 2.18

$\text{Evaluate}\phantom{\rule{0.2em}{0ex}}2{x}^{2}+3x+8\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=4.$

Evaluate:

$3{x}^{2}+4x+1\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=3.$

Evaluate:

$6{x}^{2}-4x-7\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}x=2.$

### Identify Terms, Coefficients, and 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 $7,y,5{x}^{2},9a,\text{and}\phantom{\rule{0.2em}{0ex}}13xy.$

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 2.5 gives the coefficients for each of the terms in the left column.

Term | Coefficient |
---|---|

$9a$ | $9$ |

$y$ | $1$ |

$5{x}^{2}$ | $5$ |

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 2.6 gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

Expression | Terms |
---|---|

$7$ | $7$ |

$y$ | $y$ |

$x+7$ | $x,7$ |

$2x+7y+4$ | $2x,7y,4$ |

$3{x}^{2}+4{x}^{2}+5y+3$ | $3{x}^{2},4{x}^{2},5y,3$ |

### Example 2.19

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

Identify all terms in the given expression, and their coefficients:

$4x+3b+2$

Identify all terms in the given expression, and their coefficients:

$9a+13{a}^{2}+{a}^{3}$

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

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 ${n}^{2}$ and $9{n}^{2}$ both have ${n}^{2}.$

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms $5x,7,{n}^{2},4,3x,9{n}^{2},$

### Like Terms

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

### Example 2.20

Identify the like terms:

- ⓐ$\phantom{\rule{0.2em}{0ex}}{y}^{3},7{x}^{2},14,23,4{y}^{3},9x,5{x}^{2}$
- ⓑ$\phantom{\rule{0.2em}{0ex}}4{x}^{2}+2x+5{x}^{2}+6x+40x+8xy$

Identify the like terms in the list or the expression:

$9,2{x}^{3},{y}^{2},8{x}^{3},15,9y,11{y}^{2}$

Identify the like terms in the list or the expression:

$4{x}^{3}+8{x}^{2}+19+3{x}^{2}+24+6{x}^{3}$

### Simplify Expressions by Combining Like Terms

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

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

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 $3x+6x$ 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.

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

### How To

#### Combine like terms.

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

### Example 2.21

Simplify the expression: $3x+7+4x+5.$

Simplify:

$7x+9+9x+8$

Simplify:

$5y+2+8y+4y+5$

### Example 2.22

Simplify the expression: $7{x}^{2}+8x+{x}^{2}+4x.$

Simplify:

$3{x}^{2}+9x+{x}^{2}+5x$

Simplify:

$11{y}^{2}+8y+{y}^{2}+7y$

### Translate Words to Algebraic Expressions

In the previous section, we listed many operation symbols that are 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. They are summarized in Table 2.7.

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 $a$ |
$a\xf7b$, $a/b$, $\frac{a}{b}$, $b\overline{)a}$ |

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

**to find the numbers.**

*and*### Example 2.23

Translate each word phrase into an algebraic expression:

- ⓐ the difference of $20$ and $4$
- ⓑ the quotient of $10x$ and $3$

Translate the given word phrase into an algebraic expression:

- ⓐ the difference of $47$ and $41$
- ⓑ the quotient of $5x$ and $2$

Translate the given word phrase into an algebraic expression:

- ⓐ the sum of $17$ and $19$
- ⓑ the product of $7$ and $x$

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 2.24

Translate each word phrase into an algebraic expression:

- ⓐ Eight more than $y$
- ⓑ Seven less than $9z$

Translate each word phrase into an algebraic expression:

- ⓐ Eleven more than $x$
- ⓑ Fourteen less than $11a$

Translate each word phrase into an algebraic expression:

- ⓐ$19$ more than $j$
- ⓑ $21$ less than $2x$

### Example 2.25

Translate each word phrase into an algebraic expression:

- ⓐ five times the sum of $m$ and $n$
- ⓑ the sum of five times $m$ and $n$

Translate the word phrase into an algebraic expression:

- ⓐ four times the sum of $p$ and $q$
- ⓑ the sum of four times $p$ and $q$

Translate the word phrase into an algebraic expression:

- ⓐ the difference of two times $x\phantom{\rule{0.2em}{0ex}}\text{and 8}\phantom{\rule{0.2em}{0ex}}$
- ⓑ two times the difference of $x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8$

Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to 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 2.26

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.

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.

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

### Example 2.27

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.

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.

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.

### Media Access Additional Online Resources

### Section 2.2 Exercises

#### Practice Makes Perfect

**Evaluate Algebraic Expressions**

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

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

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

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

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

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

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

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

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

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

**Identify Terms, Coefficients, and Like Terms**

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

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

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

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

$13m$

$6{x}^{3}$

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

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

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

**Simplify Expressions by Combining Like Terms**

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

$15x+4x$

$18z+9z$

$6y+4y+y$

$8a+5a+9$

$8d+6+2d+5$

$8x+7+4x-5$

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

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

**Translate English Phrases into Algebraic Expressions**

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

The sum of 9 and 1

8 less than 19

The product of 8 and 7

The quotient of 42 and 7

$3$ less than $x$

The product of $9$ and $y$

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

The quotient of $y$ and $8$

Seven times the difference of $y$ and one

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.

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.

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.

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?

**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 $\text{\u201c4}$ times the sum of $x$ and $y\text{\u201d}$ and “the sum of $4$ times $x$ and $y\text{.\u201d}$

#### 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?