# 2.3.2: Two-Step Equations from Verbal Models

- Page ID
- 4372

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Two-Step Equations from Verbal Models

Kara works as a babysitter for the neighborhood she lives in. She is saving money to plant an amazing sustainable home garden. For each babysitting job she took on, Kara charged $4 for bus fare plus an additional $8 for each hour she worked. On Saturday, Kara earned $26 for the entire babysitting job.

Write an equation to represent this situation, where h is the total number of hours that Kara worked.

In this concept, you will learn to write two-step equations from verbal models.

**Writing Two-Step Equations from Verbal Models**

You solve equations like 3x=5,(x/−2)=−4, and x+31=8 by doing one operation to both sides of the equation to isolate the variable.

But how can you solve the following equations?

3x+5=20

(x/3)−2=5

In these equations you need to do two operations to each side of the equation to isolate the variable. However, before you do this, look at some examples of word problems that involve two steps.

Here is an example.

Change the following word problem into an equation.

Six times a number, plus five is forty-one.

First, change the language into number and symbols. “Times” means multiplication, “a number” since it is not identified is your variable x, “plus” means addition, and the word “is” means equals.

The answer is 6x+5=41.

Here is another example.

Change the following word problem into an equation.

Four less than two times a number is equal to eight.

First, change the language into an equation. “Less than” means subtraction but be careful about the order. “Times” means multiplication, “a number” is your variable x, and “is equal to” means the same thing as equals.

The answer is 2x−4=8.

**Examples**

Example 2.3.2.1

Earlier, you were given a problem about Kara’s sustainable garden.

She is saving money to build one at home. She has a babysitting job where she earns $8 and hour, but she also charges $4 for bus fare. If she earned $26 in total, can you write and equation to represent this.

**Solution**

First, let h be the number of hours Kara worked.

Next, re-phrase the text to make it easier to understand. Her total earnings were $4 plus the number of hours she worked times 8. This equals $26 the total amount earned.

Then, turn the language into numbers and symbols and write the equation. “Plus” means addition, “the number of hours” is the variable h, “times” means multiplication.

4+8h=26

The answer is 4+8h=26.

Example 2.3.2.2

Write an equation for this statement.

A number divided by two, and then added to six is equal to fourteen.

**Solution**

First, change the language into numbers and mathematical symbols. “A number” is your variable x, “divided by” means division, “and then added to” means that after you divide you add, and “is” means equals.

Then, write the equation.

(x/2)+6=14

The answer is (x/2)+6=14.

**Write an equation for each word problem.**

Example 2.3.2.3

The product of five and a number, plus three is twenty-three.

**Solution**

First, translate the language into numbers and symbols. “The product of” means multiply what is in the parenthetical expression “five and a number,” “a number” is your variable x, “plus” means addition, and “is” means equals.

Then, write your equation.

5x+3=23

The answer is 5x+3=23.

Example 2.3.2.4

Six times a number, minus four is thirty-two.

**Solution**

First, translate the language into numbers and symbols. “Times” means multiplication, “a number” is the variable x, “minus” means subtraction, and “is” means equals.

6x−4=32

The answer is 6x−4=32.

Example 2.3.2.5

A number y, divided by 3, and then added to seven is ten.

**Solution**

First, translate the language into numbers and symbols. “A number y” is your variable y, “and then added to” means you divide and then add, and “is” means equals.

(y/3)+7=10

The answer is (y/3)+7=10.

**Review**

Write each statement as two-step equations.

- Two times a number, plus seven is nineteen.
- Three times a number, and five is twenty.
- Six times a number, and ten is forty-six.
- Seven less than two times a number is twenty-one.
- Eight less than three times a number is sixteen.
- A number divided by two, plus seven is ten.
- A number divided by three, and six is eleven.
- Two less than a number divided by four is ten.
- Four times a number, and eight is twenty.
- Five times a number, take away three is twelve.
- Two times a number, and seven is twenty-nine.
- Four times a number, and two is twenty-six.
- Negative three times a number, take a way four is equal to negative ten.
- Negative two times a number, and eight is equal to negative twelve.
- Negative five times a number, minus eight is equal to seventeen.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.11.

### Vocabulary

Term | Definition |
---|---|

One-Step Equation |
A one-step equation is an algebraic equation with one operation in it that requires one step to solve. |

Two-Step Equation |
A two-step equation is an algebraic equation with two operations in it that requires two steps to solve. |

### Additional Resources

Video:

Practice: **Two-Step Equations from Verbal Models**