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# 7.10: Single Variable Division Equation

Difficulty Level: At Grade Created by: CK-12

The incoming class at tennis camp is large this year. This is a special program that will have 6 people in each class. However, there is a maximum number of 42 classes. The incoming people need to be divided into groups of 6, so that the number of groups is 42. What is the maximum capacity, c\begin{align*}c\end{align*}, of the incoming class? How can you write an equation for c\begin{align*}c\end{align*}, and then solve this equation?

In this concept, you will learn to solve single variable division equations.

### Guidance

To solve an equation in which a variable is divided by a number, you use the inverse of division, multiplication, to isolate the variable and solve the equation.

You can multiply both sides of an equation by a number because of the Multiplication Property of Equality, which states:

if a=b\begin{align*}a = b\end{align*}, then a×c=b×c\begin{align*}a \times c = b \times c\end{align*}.

This means that if you multiply one side of an equation by a number c\begin{align*}c\end{align*}, you must multiply the other side of the equation by that same number c\begin{align*}c\end{align*}, to keep the values on both sides of the equation equal.

Here is an example.

Solve the equation for k\begin{align*}k\end{align*}.

k4=12

First, use the multiplication property of equality, and multiply both sides of the equation by -4 to isolate the variable k\begin{align*}k\end{align*}.

k44×k44k4===124×1248

Next, separate the fraction and simplify.

44k1kk===484848

The answer is k=48\begin{align*}k = -48\end{align*}.

Here is another example.

Solve for n\begin{align*}n\end{align*} in the equation n1.5=10\begin{align*}\frac{n}{1.5}=10\end{align*}.

First, use the multiplication property of equality to multiply both sides of the equation by 1.5.

1.5×n1.5=10×1.5
Next, since 1.5n1.5=1.51×n1.5\begin{align*}1.5 \frac{n}{1.5} = \frac{1.5}{1} \times \frac{n}{1.5}\end{align*}, you can rewrite that multiplication as one fraction.

1.5n1.5=10×1.5

Next, you separate the fraction and simplify.

1.51.5n1nn===151515
The answer is n=15\begin{align*}n=15\end{align*}.

### Guided Practice

Three friends evenly split the total cost of the bill for their lunch. The amount each friend paid was 4.25. 1. Write a division equation to represent c\begin{align*}c\end{align*}, the total cost, in dollars, of the bill for lunch. 2. Solve the equation to solve for the total cost of the bill. Consider part a first. First, rephrase the question to help you solve the problem: The total cost, c\begin{align*}c\end{align*}, divided by three equals 4.25, the amount each person paid. Then, express this as an equation. c3=4.25 Now consider part b. Solve the equation by using the multiplication property of equality. Multiply both sides of the equation by 3. c33×c3==4.253×4.25 Next, rearrange the multiplication of fractions. 33c=12.75 Now, simplify and solve. 1cc==12.7512.75 The answer is that the bill was12.75.

### Examples

Solve each equation.

#### Example 1

x2=5

First, use the multiplication property of equality and multiply both sides of the equation by -2.

2x2=2×5

Next, simplify and solve for x\begin{align*}x\end{align*}.

22x1xx===101010

The answer is x=10\begin{align*}x = -10\end{align*}.

#### Example 2

First, use the multiplication property of equality and multiply both sides of the equation by 5.

Next, simplify and solve for \begin{align*}y\end{align*}.

The answer is \begin{align*}y = 30\end{align*}.

#### Example 3

First, use the multiplication property of equality and multiply both sides of the equation by -4.

Next, simplify and solve for \begin{align*}b\end{align*}.

The answer is \begin{align*}b = 12\end{align*}.

Remember the tennis camp? The incoming students need to be divided into groups of 6, but there can only be 42 classes in total. Can you write a division equation, where \begin{align*}c\end{align*}, is the maximum number of people in the incoming class so that there are 6 people in each group, and then solve it?

First, translate the language into an equation. Let \begin{align*}c\end{align*}, be the maximum number of people in the incoming class. This number, divided by 6, should equal 42 classes.

Next, use the multiplication property of equality and multiply both sides of the equation by 6.

Then, re-write the multiplication by a fraction and simplify.

The answer is that there can be a maximum number of 252 students in the incoming class.

### Explore More

Solve each single variable division equation for the missing value.

1. \begin{align*}\frac{x}{5} = 2\end{align*}
2. \begin{align*}\frac{y}{7} = 3\end{align*}
3. \begin{align*}\frac{b}{9} = -4\end{align*}
4. \begin{align*}\frac{b}{8} = -10\end{align*}
5. \begin{align*}\frac{b}{8} = 20\end{align*}
6. \begin{align*}\frac{x}{-3} =10\end{align*}
7. \begin{align*}\frac{y}{18} = -20\end{align*}
8. \begin{align*}\frac{a}{-9} = -9\end{align*}
9. \begin{align*}\frac{x}{11} = -12\end{align*}
10. \begin{align*}\frac{x}{3} = -3\end{align*}
11. \begin{align*}\frac{x}{5} = -8\end{align*}
12. \begin{align*}\frac{x}{1.3} = 3\end{align*}
13. \begin{align*}\frac{x}{2.4} = 4\end{align*}
14. \begin{align*}\frac{x}{6} = 1.2\end{align*}
15. \begin{align*}\frac{y}{1.5} = 3\end{align*}

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.10.

1. [1]^ License: CC BY-NC 3.0
2. [2]^ License: CC BY-NC 3.0

## Date Created:

Sep 23, 2015

Jan 26, 2016
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