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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\begin{align*}\frac{k}{-4} = 12\end{align*}

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\begin{align*}\frac{x}{-2}=5\end{align*}

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

y5=6\begin{align*}\frac{y}{5}=6\end{align*}

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

5y5=5×6

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

55y1yy===303030

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

#### Example 3

b4=3\begin{align*}\frac{b}{-4}=-3\end{align*}

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

4b4=4×3

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

44b1bb===121212

The answer is b=12\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 c\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 c\begin{align*}c\end{align*}, be the maximum number of people in the incoming class. This number, divided by 6, should equal 42 classes.

c6=42

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

6×c6=6×42

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

61×c666×c11cc====252252252252

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. x5=2\begin{align*}\frac{x}{5} = 2\end{align*}
2. y7=3\begin{align*}\frac{y}{7} = 3\end{align*}
3. b9=4\begin{align*}\frac{b}{9} = -4\end{align*}
4. b8=10\begin{align*}\frac{b}{8} = -10\end{align*}
5. b8=20\begin{align*}\frac{b}{8} = 20\end{align*}
6. x3=10\begin{align*}\frac{x}{-3} =10\end{align*}
7. y18=20\begin{align*}\frac{y}{18} = -20\end{align*}
8. a9=9\begin{align*}\frac{a}{-9} = -9\end{align*}
9. x11=12\begin{align*}\frac{x}{11} = -12\end{align*}
10. x3=3\begin{align*}\frac{x}{3} = -3\end{align*}
11. x5=8\begin{align*}\frac{x}{5} = -8\end{align*}
12. x1.3=3\begin{align*}\frac{x}{1.3} = 3\end{align*}
13. x2.4=4\begin{align*}\frac{x}{2.4} = 4\end{align*}
14. x6=1.2\begin{align*}\frac{x}{6} = 1.2\end{align*}
15. y1.5=3\begin{align*}\frac{y}{1.5} = 3\end{align*}

## Date Created:

Today, 16:26

Today, 16:26
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