7.11: Two-Step Equations from Verbal Models
Have you ever tried to earn money babysitting?
Kara worked on earning money for her trip to Boston. For each babysitting job she took on, Kara charged$2 for bus fare plus an additional $6 for each hour she worked. On Saturday, Kara earned $26 for the one babysitting job.
Write an equation to represent \begin{align*}h\end{align*}, the total number of hours that Manuel babysat during his $26 job.
Do you know how to do this?
This Concept will teach you how to write two -step equations from verbal models. You will know how to do this by the end of the Concept.
Guidance
In the last few Concepts, you worked on solving single variable equations. Here is a single variable equation.
Here is a problem like the ones you worked on in the last two Concepts.
\begin{align*}3x=15\end{align*}
Now if you think about this equation, you only had to perform one operation to solve it. We used an inverse or opposite operation to solve it. Because this is a multiplication equation, we used division to solve for the unknown variable. We could say that we performed one-step to solve this equation. The one-step was to divide. We call an equation where one operation is needed to solve for the unknown variable a one-step equation.
What happens when there is more than one operation needed to solve an equation?
When this happens, we have two-step equation. A two-step equation has more than one operation in it. Let’s look at a few two-step equations.
\begin{align*}3x+5 &= 20\\ \frac{x}{3}-2 &= 5\end{align*}
If you look at both of these equations, you will see that there are two operations present in each. The first equation has multiplication and addition. The second equation has division and subtraction. You will see other variations of two-step equations too, but this gives you an idea of a few different situations.
Before we begin solving equations, let’s look at how we can write a two-step equation from a verbal model.
First, let’s think about some of the words that mean addition, subtraction, multiplication and division. Identifying these key words is going to assist us when writing equations from verbal models. Remember that a verbal model uses words.
Addition | Subtraction | Multiplication | Division |
---|---|---|---|
Sum Added Altogether In all Plus and |
Difference Less than More than Take away subtract |
Product Times Groups of |
Quotient Split Up Divided |
Take a few minutes to write down these key words in your notebook.
Now apply this information.
Six times a number plus five is forty-one.
First, identify any key words that identify operations.
Six times a number plus five is forty-one.
Next, begin to translate the words into an equation.
Six = 6
Times \begin{align*}=x\end{align*}
A number = variable
Plus = +
Five = 5
Is means =
Forty-one is 41
Next, we put it altogether.
\begin{align*}6x+5=41\end{align*}
Here is another one.
Four less than two times a number is equal to eight.
First, identify any key words that identify operations.
Four less than two times a number is equal to eight.
Now we can translate each part.
Four becomes 4
Less than means subtraction
Two becomes 2
Times \begin{align*}= x\end{align*}
A number = variable
Is equal to means =
Eight = 8
Notice that the one tricky part is in the words “less than” because it is less than two times a number, the two times a number needs to come first in the equation. Then we can subtract.
Put it altogether.
\begin{align*}2x-4=8\end{align*}
Now it's time for you to try a few on your own. Write an equation for each situation.
Example A
The product of five and a number plus three is twenty-three.
Solution:\begin{align*}5x+3=23\end{align*}
Example B
Six times a number minus four is thirty-two.
Solution:\begin{align*}6y-4=32\end{align*}
Example C
A number, \begin{align*}y\end{align*}, divided by 3 plus seven is ten.
Solution:\begin{align*}\frac{y}{3}+7=10\end{align*}
Here is the original problem once again.
Kara worked on earning money for her trip to Boston. For each babysitting job she took on, Kara charged $2 for bus fare plus an additional $6 for each hour she worked. On Saturday, Kara earned $26 for the one babysitting job.
Write an equation to represent \begin{align*}h\end{align*}, the total number of hours that Manuel babysat during his $26 job.
Do you know how to do this?
To begin, Kara worked an unknown amount of hours. This is our variable. We can call this \begin{align*}h\end{align*}
She earns six dollars an hour. We multiply that times the unknown number of hours.
\begin{align*}6h\end{align*}
Kara also charges $2.00 for bus fare.
\begin{align*}6h + 2\end{align*}
She earned twenty -six dollars.
\begin{align*}6h + 2 = 26\end{align*}
This is our equation.
Vocabulary
Here are the vocabulary words in this Concept.
- One-Step Equation
- an equation with one operation in it
- Two-Step Equation
- an equation with two operations in it
Guided Practice
Here is one for you to try on your own.
Write an equation for this statement.
A number divided by two and six is equal to fourteen.
Answer
First, look for key words that identify operations.
A number divided by two and six is equal to fourteen.
Next, translate each word into an equation.
A number means variable
Divided means \begin{align*}\div\end{align*}
By two means 2 is our divisor
And means addition
Six means 6
Is means =
Fourteen means 14
Put it altogether.
\begin{align*}\frac{x}{2}+6=14\end{align*}
Practice
Directions: Write the following two-step equations from verbal models.
1. Two times a number plus seven is nineteen.
2. Three times a number and five is twenty.
3. Six times a number and ten is forty-six.
4. Seven less than two times a number is twenty-one.
5. Eight less than three times a number is sixteen.
6. A number divided by two plus seven is ten.
7. A number divided by three and six is eleven.
8. Two less than a number divided by four is ten.
9. Four times a number and eight is twenty.
10. Five times a number take away three is twelve.
11. Two times a number and seven is twenty - nine.
12. Four times a number and two is twenty- six.
13. Negative three time a number take a way four is equal to negative ten.
14. Negative two times a number and eight is equal to negative twelve.
15. Negative five times a number minus eight is equal to seventeen.
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Here you'll learn to write two-step equations from verbal models.