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# Single Variable Expressions

## Use symbols and operations to understand and define variables.

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Single Variable Expressions

Ayako just got her first job at her local arcade. In her interview, the manager said she would earn either $8 or$9 an hour. When he called to offer her the job, Ayako was so excited that she forgot to ask what she would be paid. Ayako plans to work 20 hours per week. How can Ayako figure out how much she will get paid each week?

In this concept, you will learn how to evaluate single variable expressions with given values.

### Guidance

A variable is a letter that is used to represent an unknown quantity. Often, x\begin{align*}x\end{align*} or y\begin{align*}y\end{align*} represents the unknown quantity, but any letter, usually lowercase, can be used as a variable.

A variable can be used in any sort of mathematical expressionA variable expression is an expression with one or more operations that has variables but no equals sign.

To understand variable expressions a little better, let’s think about some ways that you can show addition, subtraction, multiplication and division in mathematics.

Addition can be shown by using a + sign.

Subtraction can be shown using a subtraction or minus sign \begin{align*}-\end{align*}.

Multiplication can be shown in different ways:

• Using the times symbol, e.g. 5×6=30.\begin{align*}5 \times 6 = 30.\end{align*}
• Using two sets of parentheses, e.g. (5)(6)=30\begin{align*}(5)(6) = 30\end{align*}
• Using a variable next to a number, e.g. 6x\begin{align*}6x\end{align*} indicates 6 multiplied by the unknown x\begin{align*}x\end{align*}
• Using one number next to parentheses, e.g. 4(3) = 12

Division can be shown in different ways:

• Using the division symbol, e.g. 6÷2=3\begin{align*} 6 \div 2 =3\end{align*}
• Using the fraction bar, e.g. 62=3\begin{align*}\frac{6}{2}=3\end{align*}

Depending on the situation, you should expect to see operations shown in different ways.

Often you will be asked to evaluate variable expressions with a given value for the variable.

Evaluate 5 + a when a = 18.

First, substitute the given value for the variable. In this case, substitute 18 in for a.

5+a5+18Substitute 18 in place of a\begin{align*}5 + a & \quad \text{Substitute 18 in place of }a\\ 5 + 18 &\\\end{align*}

5+1823Add 5+18\begin{align*}5+18 & \quad \text{Add }5+18\\ 23 &\\ \end{align*}

Let's look at an example.

Evaluate b22\begin{align*}b-22\end{align*} when b\begin{align*}b\end{align*} is 40.

First, substitute the given value for the variable. In this case, substitute 40 in for b\begin{align*}b\end{align*}

b224022Substitute 40 in place of b\begin{align*}b-22 & \quad \text{Substitute 40 in place of }b\\ 40 - 22 &\\\end{align*}

Then subtract.

402218Subtract 4022\begin{align*}40 - 22 & \quad \text{Subtract }40-22\\ 18 &\\\end{align*}

Let's look at another example.

Evaluate 7x\begin{align*}7x\end{align*} when x\begin{align*}x\end{align*} is 12.

First, substitute the given value for the variable. In this case, substitute 12 in for x\begin{align*}x\end{align*}. Then multiply.

7x7(12)84Substitute 12 in for xMultiply 7×12\begin{align*}7x & \quad \text{Substitute 12 in for }x\\ 7(12) & \quad \text{Multiply }7 \times 12\\ 84 & \\\end{align*}

Let's look at one more example.

Evaluate 14x\begin{align*}\frac{14}{x}\end{align*} when x\begin{align*}x\end{align*} is 2.

This expression includes a fraction bar, which tells you that this is a division problem.

First, substitute the given value of 2 in for \begin{align*}x\end{align*}.

\begin{align*}\frac{14}{x} & \quad \text{Substitute 2 in place of }x\\ \frac{14}{2} & = 7\end{align*}

### Examples

#### Example 1

Evaluate \begin{align*}17+y\end{align*} when \begin{align*}y\end{align*} is 12.

\begin{align*}17 + y & \quad \text{Substitute 12 in place of }y\\ 17 + 12 & \quad \text{Add } 17 + 12 = 29\\ 29 &\\\end{align*}

#### Example 2

Evaluate \begin{align*}5c\end{align*} when \begin{align*}c\end{align*} is 9.

\begin{align*}5c & \quad \text{Substitute 9 in place of }c\\ 5(9) & \quad \text{Multiply }5 \times 9\\ 45 &\\\end{align*}

#### Example 3

Evaluate \begin{align*}8 \div x\end{align*} when \begin{align*}x\end{align*} is 4.

\begin{align*}8 \div x & \quad \text{Substitute 4 in place of }x\\ 8 \div 4 & \quad \text{Divide 8 by 4}\\ 2\end{align*}

Remember Ayako and her new job?

Ayako needs to figure out how much she will get paid for working 20 hours a week if she gets paid $8 or$9 an hour.

First, write an expression.

\begin{align*}20x =\end{align*} ?

The given values for the unknown variable, \begin{align*}x\end{align*}, are 8 and 9.

Next, substitute 8 in for the variable.

20(8)=160

Then, substitute 9 in for the variable.

20(9)=180

The answer is 160 or 180.

Ayako will get paid $160 or$180 per week at her new arcade job.

### Explore More

Evaluate each of the variable expressions when \begin{align*}a = 4, \ b = 5, \ \text{and }c = 6\end{align*}

1.  \begin{align*}5 + a\end{align*}

2.  \begin{align*}6 + b\end{align*}

3.  \begin{align*}7 + c\end{align*}

4.  \begin{align*}8 - a\end{align*}

5.  \begin{align*}9c\end{align*}

6.  \begin{align*}10a\end{align*}

7.  \begin{align*}7c\end{align*}

8.  \begin{align*}9a\end{align*}

9.  \begin{align*}4b\end{align*}

10.  \begin{align*}\frac{16}{a}\end{align*}

11.  \begin{align*}\frac{42}{c}\end{align*}

12.  \begin{align*}\frac{c}{2}\end{align*}

13.  \begin{align*}15a\end{align*}

14.  \begin{align*}9b\end{align*}

15.  \begin{align*}\frac{15}{b}\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.13.

### Vocabulary Language: English

Algebraic Expression

Algebraic Expression

An expression that has numbers, operations and variables, but no equals sign.
Evaluate

Evaluate

To evaluate an expression or equation means to perform the included operations, commonly in order to find a specific value.
Expression

Expression

An expression is a mathematical phrase containing variables, operations and/or numbers. Expressions do not include comparative operators such as equal signs or inequality symbols.
Variable Expression

Variable Expression

A variable expression is a mathematical phrase that contains at least one variable or unknown quantity.