CHAPTER 1 Whole Numbers, Integers, and Introduction to Algebra

1.3 Evaluate, Simplify, and Translate Expressions

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

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 1

Evaluate when

1.
2.
Solution

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

When , the expression has a value of .

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

When , the expression has a value of .

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 were different. When we evaluate an expression, the value varies depending on the value used for the variable.

TRY IT 1.1

Evaluate:

1.
2.
1.  10
2.  19

TRY IT 1.2

Evaluate:

1.
2.
1.  4
2.  12

EXAMPLE 2

Evaluate

1.
2.
Solution

Remember means times , so means times .

a. To evaluate the expression when , we substitute for , and then simplify.

 Multiply. Subtract.

b. To evaluate the expression when , we substitute for , and then simplify.

 Multiply. Subtract.

Notice that in part a) that we wrote and in part b) we wrote . Both the dot and the parentheses tell us to multiply.

TRY IT 2.1

Evaluate:

1.  13
2.  5

TRY IT 2.2

Evaluate:

1.  8
2.  16

EXAMPLE 3

Evaluate when .

Solution

We substitute for , and then simplify the expression.

 Use the definition of exponent. Multiply.

When , the expression has a value of .

TRY IT 3.1

Evaluate:

.

64

TRY IT 3.2

Evaluate:

.

216

EXAMPLE 4

.

Solution

In this expression, the variable is an exponent.

 Use the definition of exponent. Multiply.

When , the expression has a value of .

TRY IT 4.1

Evaluate:

.

64

TRY IT 4.2

Evaluate:

.

81

EXAMPLE 5

.

Solution

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

 Multiply. Add and subtract left to right.

When and , the expression has a value of .

TRY IT 5.1

Evaluate:

33

TRY IT 5.2

Evaluate:

10

EXAMPLE 6

.

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, means and is different from the expression , which means .

TRY IT 6.1

Evaluate:

.

40

TRY IT 6.2

Evaluate:

.

9

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 .

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 is . When we write , the coefficient is , since . The table below gives the coefficients for each of the terms in the left column.

Term Coefficient

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

Expression Terms

EXAMPLE 7

Identify each term in the expression . Then identify the coefficient of each term.

Solution

The expression has four terms. They are , and .

The coefficient of is .

The coefficient of is .

Remember that if no number is written before a variable, the coefficient is . So the coefficient of is .

The coefficient of a constant is the constant, so the coefficient of is .

TRY IT 7.1

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

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

TRY IT 7.2

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

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?

Which of these terms are like terms?

• The terms and are both constant terms.
• The terms and are both terms with .
• The terms and both have .

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms ,

Like Terms

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

EXAMPLE 8

Identify the like terms:

Solution

a.

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

The terms and are like terms because they both have .

The terms and are like terms because they both have .

The terms and are like terms because they are both constants.

The term does not have any like terms in this list since no other terms have the variable raised to the power of .

b.

Look at the variables and exponents. The expression contains the terms

The terms and are like terms because they both have .

The terms are like terms because they all have .

The term has no like terms in the given expression because no other terms contain the two variables .

TRY IT 8.1

Identify the like terms in the list or the expression:

9, 15; 2x3 and 8x3, y2, and 11y2

TRY IT 8.2

Identify the like terms in the list or the expression:

4x3 and 6x3; 8x2 and 3x2; 19 and 24

Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think would simplify to? If you thought , 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 is. If you have of something and add more of the same thing, the result is of them. For example, oranges plus oranges is oranges. We will discuss the mathematical properties behind this later.

The expression 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

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

EXAMPLE 9

Simplify the expression: .

Solution
 Identify the like terms. Rearrange the expression, so the like terms are together. Add the coefficients of the like terms. The original expression is simplified to…

TRY IT 9.1

Simplify:

16x + 17

TRY IT 9.2

Simplify:

17y + 7

EXAMPLE 10

Simplify the expression: .

Solution
 Identify the like terms. Rearrange the expression so like terms are together. Add the coefficients of the like terms.

These are not like terms and cannot be combined. So is in simplest form.

TRY IT 10.1

Simplify:

4x2 + 14x

TRY IT 10.2

Simplify:

12y2 + 15y

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 the table below.

Operation Phrase Expression
the sum of and
increased by
more than
the total of and
Subtraction minus
the difference of and
subtracted from
decreased by
less than
Multiplication times
the product of and
, , ,
Division divided by
the quotient of and
the ratio of and
divided into
, , ,

Look closely at these phrases using the four operations:

• the sum of and
• the difference of and
• the product of and
• the quotient of and

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

EXAMPLE 11

Translate each word phrase into an algebraic expression:

1.  the difference of and
2.  the quotient of and
Solution

a. 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.

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

This can also be written as

TRY IT 11.1

Translate the given word phrase into an algebraic expression:

1.  the difference of and
2.  the quotient of and
1.  47 − 41
2.  5x ÷ 2

TRY IT 11.2

Translate the given word phrase into an algebraic expression:

1.  the sum of and
2.  the product of and
1.  17 + 19
2.  7x

How old will you be in eight years? What age is eight more years than your age now? Did you add 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 from your present age. Seven less than means seven subtracted from your present age.

EXAMPLE 12

Translate each word phrase into an algebraic expression:

1.  Eight more than
2.  Seven less than
Solution

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

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

TRY IT 12.1

Translate each word phrase into an algebraic expression:

1.  Eleven more than
2.  Fourteen less than
1. x + 11
2. 11a − 14

TRY IT 12.2

Translate each word phrase into an algebraic expression:

1. more than
2. less than
1. j + 19
2. 2x − 21

EXAMPLE 13

Translate each word phrase into an algebraic expression:

1. five times the sum of and
2. the sum of five times and
Solution

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

five times the sum of and

b. 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 and .

the sum of five times and

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

TRY IT 13.1

Translate the word phrase into an algebraic expression:

1. four times the sum of and
2. the sum of four times and
1. 4(p + q)
2. 4p + q

TRY IT 13.2

Translate the word phrase into an algebraic expression:

1. the difference of two times
2. two times the difference of
1. 2x − 8
2. 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 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 14

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

Solution
 Write a phrase about the height. less than the width Substitute for the width. less than Rewrite ‘less than’ as ‘subtracted from’. subtracted from Translate the phrase into algebra.

TRY IT 14.1

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

w − 5

TRY IT 14.2

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

l + 2

EXAMPLE 15

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

Solution
 Write a phrase about the number of dimes. two less than five times the number of quarters Substitute for the number of quarters. less than five times Translate times . less than Translate the phrase into algebra.

TRY IT 15.1

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

6q − 7

TRY IT 15.2

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

4n + 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

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.

Practice Makes Perfect

Evaluate Algebraic Expressions

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

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Identify Terms, Coefficients, and Like Terms

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

 21 22 23 24

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

 25 26 27 28

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

 29 30 31 32

Simplify Expressions by Combining Like Terms

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

 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Translate English Phrases into Algebraic Expressions

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

 49. The sum of 8 and 12 50. The sum of 9 and 1 51. The difference of 14 and 9 52. 8 less than 19 53. The product of 9 and 7 54. The product of 8 and 7 55. The quotient of 36 and 9 56. The quotient of 42 and 7 57. The difference of and 58. less than 59. The product of and 60. The product of and 61. The sum of and 62. The sum of and 63. The quotient of and 64. The quotient of and 65. Eight times the difference of and nine 66. Seven times the difference of and one 67. Five times the sum of and 68.  times five less than twice

In the following exercises, write an algebraic expression.