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# 1.12: Solve Real-World Problems by Writing and Solving Single-Variable Equations

Difficulty Level: At Grade Created by: CK-12
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Practice Sentences as Single Variable Equations

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Sheldon is watching his sister, Tracie, build triangles with red popsicle sticks. Tracie challenges him to tell her how many triangles she can build with 21 popsicle sticks. How can Sheldon use a table and an equation to figure this out?

In this concept, you will learn to solve real-world problems by writing and solving single-variable equations.

### Solving Single Variable Equations

Writing an equation to model a real-world problem is often easier when you take the information given in the problem and express it in verbal form by using a few key words. This is very similar to translating verbal phrases into variable expressions. The difference is you will be translating a verbal sentence into an equation. Then, you can solve the equation using inverse operations.

Let’s look at an example.

Monica purchased a pair of tennis shoes that had this sticker on the bottom of the shoe.

$99.00$65.99\begin{align*}\begin{array}{rcl} & \cancel{\ 99.00}\\ & \ 65.99 \end{array}\end{align*}

Use a verbal model to write and solve an equation to determine the amount of money Monica saved by purchasing the shoes on sale.

First, write a verbal model to represent the problem.

Verbal Model: Sales Price+Amount Saved=Original Price\begin{align*}\text{Sales Price} + \text{Amount Saved} = \text{Original Price}\end{align*}

Next, name the variable.

Let ‘s\begin{align*}s\end{align*}’ represent the amount saved.

Next, express the verbal model as an equation.

Sale Price65.99+Amount Saveds65.99+s==Original Price99.0099.00\begin{align*}\begin{array}{rcl} \underbrace{ \text{Sale Price} }_{65.99} + \underbrace{ \text{Amount Saved} }_{s} & = & \underbrace{ \text{Original Price} }_{99.00}\\ 65.99+s & = & 99.00 \end{array}\end{align*}

Next, solve the equation you have written. Remember an equation is like a balance scale. To keep the scale balanced, what you do to one side of the equation you must also do to the other side. When solving an equation your goal is to isolate the variable on one side of the equation and the numerical terms on the other side. This is done by performing inverse operations.

65.99+s=99.00\begin{align*}65.99+s=99.00\end{align*}

First, isolate the variable ‘s\begin{align*}s\end{align*}’ by subtracting 65.99 from both sides of the equation. Subtraction is the inverse of addition.

65.9965.99+s=99.0065.99\begin{align*}65.99-65.99+s=99.00-65.99\end{align*}

Next, simplify each side.

s=33.01\begin{align*}s= 33.01\end{align*}

The answer is 33.01. ### Examples #### Example 1 Earlier, you were given a problem about Sheldon’s red popsicle stick challenge. Sheldon needs to tell his sister how many triangles she can make with 21 popsicle sticks. To figure out the number of triangles, Sheldon has to write an equation to represent the pattern. First, create a table of values.  Number of Triangles (n)\begin{align*}(n)\end{align*} 1 2 3 4 ... n\begin{align*}n\end{align*} Number of popsicle sticks (p)\begin{align*}(p)\end{align*} 3 5 7 9 21 Next, write down the information in the table. For the first triangle n=1\begin{align*}n=1\end{align*} and p=3\begin{align*}p=3\end{align*}. For the second triangle n=2\begin{align*}n=2\end{align*} and p=5\begin{align*}p=5\end{align*}. For the third triangle n=3\begin{align*}n=3\end{align*} and p=7\begin{align*}p=7\end{align*}. For the fourth triangle n=4\begin{align*}n=4\end{align*} and p=9\begin{align*}p=9\end{align*}. Next, model the information by writing a verbal model. Verbal Model: Twice the number of triangles plus 1 equals 21. Next, write and solve a single- variable equation to model the problem. 2n+1=21\begin{align*}2n+1=21\end{align*} Next, isolate the variable by subtracting one from both sides of the equation. 2n+11=211\begin{align*}2n+1-1=21-1\end{align*} Next, simplify both sides of the equation. 2n=20\begin{align*}2n=20\end{align*} Then, divide both sides of the equation by 2 to solve for ‘n\begin{align*}n\end{align*}’. 12n2n==20210\begin{align*}\begin{array}{rcl} \frac{^1\cancel{2}n}{\cancel{2}} & = & \frac{20}{2}\\ n & = & 10 \end{array}\end{align*} The answer is 10. Sheldon can tell his sister she can build 10 triangles with 21 popsicle sticks. #### Example 2 Determine the number of cattle using a single-variable equation. Six times the number of cattle less 87 is 999. First, name the variable. Let ‘c\begin{align*}c\end{align*}’ represent the number of cattle. Next, express the statement as an equation. Six times the number of cattle6cless 8787is 999=9996c87=999\begin{align*}\begin{array}{rcl} && \underbrace{ \text{Six times the number of cattle} }_{6c} \quad \underbrace{ \text{less 87} }_{-87} \quad \underbrace{ \text{is 999} }_{=999}\\ && 6c-87=999 \end{array}\end{align*} Next, isolate the variable by adding 87 to both sides of the equation. Addition is the inverse of subtraction. 6c87+87=999+87\begin{align*}6c-87+87=999+87\end{align*} Next, simplify both sides of the equation. 6c=1086\begin{align*}6c=1086\end{align*} Then, divide both sides of the equation by 6 to determine the value of ‘c\begin{align*}c\end{align*}’. Division is the inverse of multiplication. 16c6c==10866181\begin{align*}\begin{array}{rcl} \frac{^1\cancel{6}c}{\cancel{6}} & = & \frac{1086}{6}\\ c & = & 181 \end{array}\end{align*} The answer is 181. #### Example 3 A safety fence around a swimming pool has a rectangular gate entrance to the pool area. If the perimeter of the triangle outlined in dark blue is 12 yards, write and solve an equation to determine the lengths of the sides of the triangle. First, name the variable. Let ‘x\begin{align*}x\end{align*}’ represent the unknown length. Next, write an equation to represent the perimeter of the triangle. Remember the perimeter is the sum of the lengths of the three sides of the outlined blue triangle. x+(x+1)+(x+2)x+x+1+x+2==1212\begin{align*}\begin{array}{rcl} x+(x+1)+(x+2) & = & 12\\ x+x+1+x+2 & = & 12 \end{array}\end{align*} Next, simplify the left side of the equation by collecting like terms. 3x+3=12\begin{align*}3x+3=12\end{align*} Next, isolate the variable by subtracting three from both sides of the equation. 3x+33=123\begin{align*}3x+3-3=12-3\end{align*} Next, simplify both sides of the equation. 3x=9\begin{align*}3x=9\end{align*} Next, divide both sides of the equation by 3 to solve for ‘x\begin{align*}x\end{align*}’. 13x3x==933\begin{align*}\begin{array}{rcl} \frac{^1\cancel{3}x}{\cancel{3}} & = & \frac{9}{3}\\ x & = & 3 \end{array}\end{align*} The answer is 3. Then, substitute x=3\begin{align*}x=3\end{align*} into the length for each of the three sides of the triangle. x+(x+1)+(x+2)3+(3+1)+(3+2)xx+1x+2=====12123 yd4 yd5 yd\begin{align*}\begin{array}{rcl} x+(x+1)+(x+2) & = & 12\\ 3+(3+1)+(3+2) & = & 12\\ x & = & 3 \ yd\\ x+1 & = & 4 \ yd\\ x+2 & = & 5 \ yd \end{array}\end{align*} #### Example 4 Jonas is working with his Uncle Tim as an apprentice electrician wiring a new house. For each job that Jonas works, he is paid a one-time service fee of50.00 plus $25.00 an hour. When the house was completely wired, Uncle Tim paid Jonas$2250.00.

Write and solve an equation to determine the number of hours Jonas worked wiring the house.

First, name the variable.

Let ‘\begin{align*}h\end{align*}’ be the number of hours Jonas worked.

Next, write a verbal model to represent the problem.

Verbal Model: 25 times the number of hours worked plus the service fee is 2250.

Next, write a single-variable equation to represent the verbal model.

\begin{align*}\underbrace{ \text{25 times the number of hours worked} }_{25h} \quad \underbrace{ \text{plus the service fee} }_{+50} \quad \underbrace{ \text{is 2250} }_{=2250}\end{align*}

The single-variable equation is

\begin{align*}25h+50=2250\end{align*}

Next, isolate the variable by subtracting 50 from both sides of the equation.

\begin{align*}25h+50-50=2250-50\end{align*}

Next, simplify both sides of the equation.

\begin{align*}25h=2200\end{align*}

Then, divide both sides of the equation by 25 to solve for ‘\begin{align*}h\end{align*}’.

\begin{align*}\begin{array}{rcl} \frac{^1\cancel{25}h}{\cancel{25}} & = & \frac{2200}{25}\\ h & = & 88 \end{array}\end{align*}

### Review

Write an equation for each situation and then solve for the variable. Each problem will have two answers.

1. An unknown number and three is equal to twelve.
2. John had a pile of golf balls. He lost nine on the course. If he returned home with fourteen golf balls, how many did he start with?
3. Some number and six is equal to thirty.
4. Jessieowes her brother some money. She earned nine dollars and paid off some of her debt. If she still owes him five dollars, how much did she owe him to begin with?
5. A farmer has chickens. Six of them went missing during a snow storm. If there are twelve chickens left, how many did he begin with before the storm?
6. Gasoline costs four dollars per gallon. Kerry put many gallons in his car over a long car trip. If he spent a total of \$140.00 on gasoline, how many gallons did he need for the trip?
7. Twenty-seven times a number is 162. What is the number?
9. The coach divided the students into five teams. There were fourteen students on each team. How many students did the coach begin with?
10. A number plus nineteen is equal to forty.

To see the Review answers, open this PDF file and look for section 1.12.

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Color Highlighted Text Notes

### Vocabulary Language: English

Algebraic Equation

An algebraic equation contains numbers, variables, operations, and an equals sign.

Consecutive

The term consecutive means "one after another." An example of consecutive numbers is 1, 2, and 3. An example of consecutive even numbers would be 2, 4, and 6. An example of consecutive odd numbers would be 1, 3, and 5.

Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.

Verbal Model

A verbal expression (or verbal model) uses words to decipher the mathematical information in a problem. An equation can often be written from a verbal model.

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