2.2 Solve Equations using the Division and Multiplication Properties of Equality

Learning objectives

Learning objectives

• Solve equations using the Division and Multiplication Properties of Equality

• Solve equations that require simplification

• Translate to an equation and solve

• Translate and solve applications

Solve Equations Using the Division and Multiplication Properties of Equality

You may have noticed that all of the equations we have solved so far have been of the form $x+a=b$ or $x-a=b$. We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.

Let’s look at our puzzle again with the envelopes and counters in Figure 2.5.

In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?

How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since $6\mathrm{÷}2=3$).

What equation models the situation shown in Figure 2.6? There are two envelopes, and each contains $x$ counters. Together, the two envelopes must contain a total of 6 counters.

We found that each envelope contains 3 counters. Does this check? We know $2\cdot{}3=6$, so it works! Three counters in each of two envelopes does equal six!

This example leads to the Division Property of Equality.

The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.

Example 2.13

Solution

Consider the equation $\frac{x}{4}=3$. We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.

Example 2.14

Solution

Here $y$ is divided by $-7$. We must multiply by $-7$ to isolate $y$.

Example 2.15

Solution

Example 2.16

Solution

Since the product of a number and its reciprocal is 1, our strategy will be to isolate $x$ by multiplying by the reciprocal of $\frac{3}{4}$.

In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.

Example 2.17

Solution

Solve Equations That Require Simplification

Many equations start out more complicated than the ones we have been working with.

With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.

Example 2.18

Solution

Begin by simplifying each side of the equation.

Example 2.19

Solution

Here we will simplify each side of the equation by using the distributive property first.

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Translate to an Equation and Solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

Example 2.20

Solution

Begin by translating the sentence into an equation.

Example 2.21

Solution

Example 2.22

Solution

Begin by translating the sentence into an equation.

Example 2.23

Solution

Begin by translating the sentence into an equation. Remember, “of” translates into multiplication.

Example 2.24

Solution

Begin by translating the sentence into an equation.

Translate and Solve Applications

To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.

Example 2.25

Solution

| | |

| --- | --- |

| What are you asked to find? | The cost of 1 pound of grapes |

| Assign a variable. | Let $c$ = the cost of one pound. |

| Write a sentence that gives the information to find it. | The cost of 6 pounds is $10.74. |

| Translate into an equation. | $6c=10.74$ |

| Solve. | $\begin{array}{l} \frac{6c}{6}=\frac{10.74}{6} \\ c=1.79 \\ \end{array}$ |

| | The grapes cost $1.79 per pound. |

| Check: If one pound costs $1.79, do 6 pounds cost \$10.74?$

| |

Table

2.3

Example 2.26

Solution

Table

2.4

Section 2.2 Exercises

Section 2.2 Exercises

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the Division and Multiplication Properties of Equality and check the solution.

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

Mixed Practice

In the following exercises, solve each equation.

Translate to an Equation and Solve

In the following exercises, translate to an equation and then solve.

Translate and Solve Applications

In the following exercises, translate into an equation and solve.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. Solve: $3y=-41.$

2. Solve: $4z=-55.$

3. Solve: $\frac{a}{-7}=-42.$

4. Solve: $\frac{b}{-6}=-24.$

5. Solve: $-k=8.$

1. Solve: $-g=3.$

2. Solve: $\frac{2}{5}n=14.$

3. Solve: $\frac{5}{6}y=15.$

4. Solve: $\frac{9}{25}=-\frac{4}{5}\,z.$

5. Solve: $\frac{5}{6}=-\frac{8}{3}\,r.$

1. Solve: $18-27=15c-9c-3c.$

2. Solve:$18-22=12x-x-4x.$

3. Solve: $-4(q-2)-8=24.$

4. Solve: $-6(r-2)-12=30.$

5. Translate and solve: The number 132 is the product of −12 and y.

1. Translate and solve: The number 117 is the product of −13 and z.

2. Translate and solve: $n$ divided by 7 is equal to $-21$.

3. Translate and solve: $n$ divided by 8 is equal to $-56$.

4. Translate and solve: The quotient of $q$ and $-8$ is 72.

5. Translate and solve: The quotient of $p$ and $-9$ is 81.

1. Translate and solve: Two-fifths of $f$ is 16.

2. Translate and solve: Three-fourths of $f$ is 21.

3. Translate and solve: The sum of five-eighths and x is one-fourth.

4. Translate and solve: The sum of three-fourths and x is five-sixths.

5. Translate and solve:

Arianna bought a 24-pack of water bottles for $9.36. What was the cost of one water bottle?

1. Translate and solve:

At JB’s Bowling Alley, 6 people can play on one lane for $34.98. What is the cost for each person?

2. Translate and solve:

The annual property tax on the Mehta’s house is $1,800, calculated as$\frac{15}{1,000}$ of the assessed value of the house. What is the assessed value of the Mehta’s house?

3. Translate and solve:

Stella planted 14 flats of flowers in $\frac{2}{3}$ of her garden. How many flats of flowers would she need to fill the whole garden?

4. $8x=56$

5. $-5c=55$

1. $-809=15y$

2. $-37p=-541$

3. $0.25z=3.25$

4. $-13x=0$

5. $\frac{x}{4}=35$

1. $-20=\frac{q}{-5}$

2. $\frac{y}{9}=-16$

3. $\frac{m}{-12}=45$

4. $-y=6$

5. $-v=-72$

1. $\frac{2}{3}y=48$

2. $-\frac{5}{8}w=40$

3. $-\frac{2}{5}=\frac{1}{10}a$

4. $-\frac{7}{10}x=-\frac{14}{3}$

5. $\frac{7}{12}=-\frac{3}{4}p$

1. $-\frac{5}{18}=-\frac{10}{9}u$

2. $100-16=4p-10p-p$

3. $\frac{7}{8}n-\frac{3}{4}n=9+2$

4. $0.25d+0.10d=6-0.75$

5. $-10(q-4)-57=93$

1. $-10(x+4)-19=85$

2. $\frac{9}{10}x=90$

3. $y+46=55$

4. $\frac{w}{-2}=99$

5. $27=6a$

1. $-x=2$

2. $m-41=-14$

3. $63.90=0.03p$

4. $84=-12z$

5. $c-0.3c=35.70$

1. $-x=-8$

2. 133 is the product of $-19$ and n.

3. $-152$ is the product of 8 and q.

4. r divided by 12 is equal to $-48$.

5. j divided by $-20$ is equal to $-80$.

1. The quotient of $b$ and $-6$ is 18.

2. The quotient $k$ and 22 is $-66$.

3. Three-tenths of x is 15.

4. Five-halves of v is 50.

5. The sum of two-fifths and f is one-half.

1. The difference of q and one-eighth is three-fourths.

2. Balloons Ramona bought 18 balloons for a party. She wants to make 3 equal bunches. How many balloons did she use in each bunch?

3. Shopping Serena paid $12.96 for a pack of 12 pairs of sport socks. What was the price of pair of sport socks?

4. MPG John’s SUV gets 18 miles per gallon (mpg). This is half as many mpg as his wife’s hybrid car. How many miles per gallon does the hybrid car get?

5. Real estate Bea earned $11,700 commission for selling a house, calculated as$\frac{6}{100}$ of the selling price. What was the selling price of the house?

1. Stamps Travis bought $9.45 worth of 49-cent stamps and 21-cent stamps. The number of 21-cent stamps was 5 less than the number of 49-cent stamps. Solve the equation$0.49s+0.21(s-5)=9.45$ for s, to find the number of 49-cent stamps Travis bought.

2. Emiliano thinks $x=40$ is the solution to the equation $\frac{1}{2}x=80$. Explain why he is wrong.