Evaluate, Simplify, and Translate Expressions

Corinn Herrell; Cheryl Colan; Fatima Baig

Evaluate, Simplify, and Translate Expressions

In this chapter, you will look at identifying expressions and terms and building and solving equations with these expressions. Many of the calculations we do in nursing are a combination of many different items that we need to combine appropriately to get the answers we seek. This lesson will create the building block for solving these problems in the various formal types we use in nursing.

Learning Objectives

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

Evaluate expressions
Identify terms, coefficients, and like terms
Simplify expressions by combining like terms
Write expressions to represent word problems

Evaluate Expressions

In this section, we’ll evaluate expressions following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when a given number replaces the variable. 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:

ⓐ $x=3$
ⓑ $x=12$

Solution:

ⓐ To evaluate, substitute the given number for $x$ in the expression, and then simplify.

Write the expression to be evaluated.	$x+7$
Substitute.	$3+7$
Add.	$10$

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

ⓑ To evaluate, substitute the given number for $x$ in the expression, and then simplify.

Write the expression to be evaluated.	$x+7$
Substitute.	$12+7$
Add.	$19$

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

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

Try It

Example 2.14

Evaluate $9x-2$ , when:

ⓐ $x=5$
ⓑ $x=1$

Solution:

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

ⓐ To evaluate, substitute the given number for $x$ in the expression, and then simplify.

Write the expression to be evaluated.	$9x-2$
Substitute $5$ for $x$ .	$9\cdot5-2$
Multiply.	$45-7$
Subtract.	$43$

ⓑ To evaluate, substitute the given number for $x$ in the expression, and then simplify.

Write the expression to be evaluated.	$9x-2$
Substitute $1$ for $x$ .	$9(1)-2$
Multiply.	$9-2$
Subtract.	$7$

Notice that in part ⓐ we wrote $9\cdot5$ , and in part ⓑ we wrote $9(1)$ . Both the dot and the parenthesis tell us to multiply.

Try It

Example 2.17

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

Solution:

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

Write the expression to be evaluated.	$3x+4y-6$
Substitute $10$ for $x$ and $2$ for $y$ .	$3(10)+4(2)-6$
Multiply.	$30+8-6$
Add and subtract left to right.	$32$

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

Try It

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 $7$ , $y$ , $5x^2$ , 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 $x=1\cdotx$ . Table 2.5 gives some example terms in the left column and the coefficients for each term in the right column.

*Table 2.5: Terms and Their Coefficients*
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 2.6 gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

*Table 2.6 Algebraic Expressions and Their Terms*
Expression	Terms
$7$	$7$
$y$	$y$
$x + 7$	$x$ , $7$
$2x + 7y + 4$	$2x$ , $7y$ , $4$
$3x^2 + 4x^2 + 5y + 3$	$3x^2$ , $4x^2$ , $5y$ , $3$

Example 2.19

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

Solution:

The expression has four terms. They are $9b$ , $15x^2$ , $a$ , and $6$ .

The coefficient of $9b$ is $9$ .

The coefficient of $15x^2$ 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 $6$ .

Try It

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

$5x, 7, n^2, 4, 3x, 9n^2$

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 $9n^2$ 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 $5x$ , $7$ , $n^2$ , $4$ , $3x$ , and $9n^2$ ,

$7$ and $4$ are like terms.
$5x$ and $3x$ are like terms.
$n^2$ and $9n^2$ are like terms.

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:

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

Solution:

ⓐ $y^3, 7x^2, 14, 23, 4y^3, 9x, 5x^2$

Look at the variables and exponents. The expression contains $y^3, x^2, x,$ and constants.

The terms $y^3$ and $4y^3$ are like terms because they both have $y^3$ .
The terms $7x^2$ and $5x^2$ are like terms because they both have $x^2$ .
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 $1$ .

ⓑ $4x^2 + 2x + 5x^2 + 6x +40x + 8xy$

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

The terms $4x^2$ and $5x^2$ are like terms because they both have $x^2$ .
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 $xy$ .

TRY IT 2.39 & 2.4O

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.

The image shows the expression 3 x plus 6 x. The 3 x represents x plus x plus x. The 6 x represents x plus x plus x plus x plus x plus x. The expression 3 x plus 6 x becomes x plus x plus x plus x plus x plus x plus x plus x plus x. This simplifies to a total of 9 x's or the term 9 x.

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 if $9$ 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.

The image shows the expression 3 x plus 4 y plus 2 x plus 6 y. The position of the middle terms, 4 y and 2 x, can be switched so that the expression becomes 3 x plus 2 x plus 4 y plus 6 y. Now the terms containing x are together and the terms containing y are together.

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

HOW TO: Combine like terms.

Steps:

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

EXAMPLE 2.21

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

Solution:

Step	Solution Example
Write out the expression.	$3x+7+4x+5$
Identify the like terms.	$3x$ and $4x$ are like terms. $7$ and $5$ are like terms.
Rearrange the expression, so the like terms are together.	$3x+4x+7+5$
Add the coefficients of the like terms.
Solution: The original expression is simplified to…	$7x+12$

TRY IT 2.41 & 2.42

EXAMPLE 2.22

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

Solution:

Step	Solution Example
Write out the expression.	$7x^2 + 8x + x^2 + 4x$
Identify the like terms.	$7x^2$ and $X^2$ are like terms. $8x$ and $4x$ are like terms.
Rearrange the expression so the like terms are together.	$7x^2 + x^2 + 8x + 4x$
Add the coefficients of the like terms.	$8x^2 + 12x$

The remaining terms are not like terms and cannot be combined. So $8x^2 + 12x$ is in simplest form.

TRY IT 2.43 AND 2.44

Write Expressions to Represent Word Problems

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.

Table 2.7: Phrases and their mathematical expressions for different operations
Operation	Phrases	Expression(s)
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 $b$	$a\cdotb, ab, a(b), (a)(b)$
Division	$a$ divided by $b$ the quotient of $a$ and $b$ the ratio of $a$ and $b$ $b$ divided into $a$	$a\divb, a/b, \dfrac{a}{b}$

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 2.23

Write an expression to represent each of these word phrases:

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

Solution:

ⓐ 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.

Phrase with words to look for emphasized: the difference of $20$ and $4$
Operation to perform: $20$ minus $4$
Written as an expression: $20-4$

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

Phrase: the quotient of $10x$ and $3$
Operation to perform: divide $10x$ by $3$
Written as an expression: $10x\div3$

This can also be written as $10x/3$ or $\dfrac{10x}{3}$ .

TRY IT 2.45 & 2.46

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

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

Example 2.24

Write an expression to represent each of these word phrases:

ⓐ eight more than $y$
ⓑ seven less than $9z$

Solution:

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

Phrase: eight more than $y$
Operation to perform: eight added to $y$
Written as an expression: $y+8$

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

Phrase: seven less than $9z$
Operation to perform: seven subtracted from $9z$
Written as an expression: $9z-7$

TRY IT 2.47 AND 2.48

EXAMPLE 2.25

Write an expression to represent each of these word phrases:

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

Solution:

ⓐ 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 $n$ .

Phrase: five times the sum of $m$ and $n$
Group the addition in parentheses: five times ( $m+n$ )
Final written expression: $5(m+n)$

ⓑ There are two operation words: sum tells us to add, and times tells us to multiply. 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 $n$ .

Phrase: the sum of five times $m$ and $n$
First, multiply five and $m$ : the sum of $5m$ and $n$
Final written expression: $5m+n$

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

TRY IT 2.49 AND 2.50

Later in this course, we’ll apply our math skills to solving equations. We’ll usually start by writing an expression to represent a word problem. We’ll need to be clear about representing the words accurately in our expressions. 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 that represents the height of the window.

Solution:

Step	Written as an expression
Write a phrase about the height.	$6$ less than the width
Substitute $w$ for the width.	$6$ less than $w$
Rewrite “less than” as “subtracted from”.	$6$ subtracted from $w$
Write a final expression representing the phrase.	$w$ – 6

TRY IT 2.51 AND 2.52

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 that represents the number of dimes.

Solution:

Step	Written as an expression
Write a phrase about the number of dimes.	$2$ less than $5$ times the number of quarters
Substitute $q$ for the number of quarters.	$2$ less than $5$ times $q$
Rewrite “ $5$ times $q$ ” as an expression	$2$ less than $5q$
Write a final expression representing the phrase.	$5q-2$

TRY IT 2.53 AND 2.54

Practice Makes Perfect

Evaluate Expressions

PRACTICE: Evaluate the expressions for the given value

Identify Terms, Coefficients, and Like Terms

PRACTICE: List the terms in the given expression.

PRACTICE: Identify the coefficient of the given terms.

PRACTICE: Identify all the sets of like terms.

Simplify Expressions by Combining Like Terms

PRACTICE: Simplify the given expression by combining like terms.

Write Expressions to Represent Word Problems

PRACTICE: Write expressions that represents the given word phrases.

PRACTICE: Write expressions for word problems.

Everyday Math

PRACTICE: In the following exercise, write an expression to represent the word problem and use it to solve.

Self-Check

After completing the exercises, use this table to evaluate your mastery of the objectives of this section. For each skill listed in a table row, rate yourself in the column that best describes your confidence level.

Expressions Self-Check
I can …	Confidently	With some help	No – I don’t get it!
evaluate expressions.
identify terms, coefficients, and like terms.
simplify expressions by combining like terms.
write expressions that represent word phrases.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident in your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, and in nursing, every topic builds upon previous work. It is important to ensure you have a strong foundation before you move on. Who can you ask for help? Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor, student success specialist, or math tutor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

Chapter Attributions

This chapter was adapted by Corinn Herrell and Cheryl Colan from “2.2 Decimals” in Prealgebra 2e by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis. Licensed under a CC BY 4.0 license.

License

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Evaluate Expressions

Identify Terms, Coefficients, and Like Terms

Simplify Expressions by Combining Like Terms

Write Expressions to Represent Word Problems

Practice Makes Perfect

Evaluate Expressions

Identify Terms, Coefficients, and Like Terms

Simplify Expressions by Combining Like Terms

Write Expressions to Represent Word Problems

Everyday Math

Self-Check

Chapter Attributions

License

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