3.5 Solve Equations Using Integers; The Division Property of Equality

Determine Whether a Number is a Solution of an Equation

In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

Solve Equations with Integers Using the Addition and Subtraction Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.

When you add or subtract the same quantity from both sides of an equation, you still have equality.

Model the Division Property of Equality

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. Now we’ll see how to solve equations that involve division.

We will model an equation with envelopes and counters in Figure 3.25.

Here, 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?

To determine the number, separate the counters on the right side into $2$ groups of the same size. So $6$ counters divided into $2$ groups means there must be $3$ counters in each group (since $6÷2=3).$

What equation models the situation shown in Figure 3.26? There are two envelopes, and each contains $x$ counters. Together, the two envelopes must contain a total of $6$ counters. So the equation that models the situation is $2x=6.$

We can divide both sides of the equation by $2$ as we did with the envelopes and counters.

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

Figure 3.27 shows another example.

Now we have $3$ identical envelopes and $\text{12 counters.}$ How many counters are in each envelope? We have to separate the $\text{12 counters}$ into $\text{3 groups.}$ Since $12÷3=4,$ there must be $\text{4 counters}$ in each envelope. See Figure 3.28.

The equation that models the situation is $3x=12.$ We can divide both sides of the equation by $3.$

Does this check? It does because $3·4=12.$

Solve Equations Using the Division Property of Equality

The previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality.

Translate to an Equation and Solve

In the past several examples, we were given an equation containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.

Key Concepts

• How to determine whether a number is a solution to an equation.
  • Step 1. Substitute the number for the variable in the equation.
  • Step 2. Simplify the expressions on both sides of the equation.
  • Step 3. Determine whether the resulting equation is true.
    • If it is true, the number is a solution.
    • If it is not true, the number is not a solution.
• Properties of Equalities

  Subtraction Property of Equality	Addition Property of Equality
  $\text{For any numbers}a,b,c,$ $\text{if}a=b\text{then}a-c=b-c.$	$\text{For any numbers}a,b,c,$ $\text{if}a=b\text{then}a+c=b+c.$

• Division Property of Equality
  • For any numbers $a,b,c,$ and $c\ne 0$
    If $a=b$, then $\frac{a}{c}=\frac{b}{c}$.

Section Exercises

Practice Makes Perfect

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

$$4x-2=6$$

1. ⓐ $x=\mathrm{-2}$
2. ⓑ $x=\mathrm{-1}$
3. ⓒ $x=2$

1. ⓐ no
2. ⓑ no
3. ⓒ yes

$$4y-10=\mathrm{-14}$$

1. ⓐ $y=\mathrm{-6}$
2. ⓑ $y=\mathrm{-1}$
3. ⓒ $y=1$

$$9a+27=\mathrm{-63}$$

1. ⓐ $a=6$
2. ⓑ $a=\mathrm{-6}$
3. ⓒ $a=\mathrm{-10}$

1. ⓐ no
2. ⓑ no
3. ⓒ yes

$$7c+42=\mathrm{-56}$$

1. ⓐ $c=2$
2. ⓑ $c=\mathrm{-2}$
3. ⓒ $c=\mathrm{-14}$

Solve Equations Using the Addition and Subtraction Properties of Equality

In the following exercises, solve for the unknown.

$$n+12=5$$

n = −7

$$m+16=2$$

$$p+9=\mathrm{-8}$$

p = −17

$$q+5=\mathrm{-6}$$

$$u-3=\mathrm{-7}$$

u = −4

$$v-7=\mathrm{-8}$$

$$h-10=\mathrm{-4}$$

h = 6

$$k-9=\mathrm{-5}$$

$$x+(\mathrm{-2})=\mathrm{-18}$$

x = −16

$$y+(\mathrm{-3})=\mathrm{-10}$$

$$r-(\mathrm{-5})=\mathrm{-9}$$

r = −14

$$s-(\mathrm{-2})=\mathrm{-11}$$

Model the Division Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve it.

3x = 6; x = 2

2x = 8; x = 4

Solve Equations Using the Division Property of Equality

In the following exercises, solve each equation using the division property of equality and check the solution.

$$5x=45$$

x = 9

$$4p=64$$

$$\mathrm{-7}c=56$$

c = −8

$$\mathrm{-9}x=54$$

$$\mathrm{-14}p=\mathrm{-42}$$

p = 3

$$\mathrm{-8}m=\mathrm{-40}$$

$$\mathrm{-120}=10q$$

q = −12

$$\mathrm{-75}=15y$$

$$24x=480$$

x = 20

$$18n=540$$

$$\mathrm{-3}z=0$$

z = 0

$$4u=0$$

Translate to an Equation and Solve

In the following exercises, translate and solve.

Four more than $n$ is equal to 1.

n + 4 = 1; n = −3

Nine more than $m$ is equal to 5.

The sum of eight and $p$ is $\mathrm{-3}$.

8 + p = −3; p = −11

The sum of two and $q$ is $\mathrm{-7}$.

The difference of $a$ and three is $\mathrm{-14}$.

a − 3 = −14; a = −11

The difference of $b$ and $5$ is $\mathrm{-2}$.

The number −42 is the product of −7 and $x$.

−42 = −7x; x = 6

The number −54 is the product of −9 and $y$.

The product of -15 and $f$ is 75.

−15f = 75; f = −5

The product of −18 and $g$ is 36.

−6 plus $c$ is equal to 4.

−6 + c = 4; c = 10

−2 plus $d$ is equal to 1.

Nine less than $m$ is −4.

m − 9 = −4; m = 5

Thirteen less than $n$ is $\mathrm{-10}$.

Mixed Practice

In the following exercises, solve.

1. ⓐ $x+2=10$
2. ⓑ $2x=10$

1. ⓐ x = 8
2. ⓑ x = 5

1. ⓐ $y+6=12$
2. ⓑ $6y=12$

1. ⓐ $\mathrm{-3}p=27$
2. ⓑ $p-3=27$

1. ⓐ p = −9
2. ⓑ p = 30

1. ⓐ $\mathrm{-2}q=34$
2. ⓑ $q-2=34$

$$a-4=16$$

a = 20

$$b-1=11$$

$$\mathrm{-8}m=\mathrm{-56}$$

m = 7

$$\mathrm{-6}n=\mathrm{-48}$$

$$\mathrm{-39}=u+13$$

u = −52

$$\mathrm{-100}=v+25$$

$$11r=\mathrm{-99}$$

r = −9

$$15s=\mathrm{-300}$$

$$100=20d$$

d = 5

$$250=25n$$

$$\mathrm{-49}=x-7$$

x = −42

$$64=y-4$$

Self Check

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

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Chapter Review Exercises

Introduction to Integers

Locate Positive and Negative Numbers on the Number Line

In the following exercises, locate and label the integer on the number line.

$$5$$

$$\mathrm{-5}$$

$$\mathrm{-3}$$

$$3$$

$$\mathrm{-8}$$

$$\mathrm{-7}$$

Order Positive and Negative Numbers

In the following exercises, order each of the following pairs of numbers, using $<$ or $\text{>.}$

$$4\text{\_\_}8$$

<

$$\mathrm{-6}\text{\_\_}3$$

$$\mathrm{-5}\text{\_\_}\mathrm{-10}$$

>

$$\mathrm{-9}\text{\_\_}\mathrm{-4}$$

$$2\text{\_\_}\mathrm{-7}$$

>

$$\mathrm{-3}\text{\_\_}1$$

Find Opposites

In the following exercises, find the opposite of each number.

$$6$$

−6

$$\mathrm{-2}$$

$$\mathrm{-4}$$

4

$$3$$

In the following exercises, simplify.

1. ⓐ $-(8)$
2. ⓑ $-(\mathrm{-8})$

1. ⓐ −8
2. ⓑ 8

1. ⓐ $-(9)$
2. ⓑ $-(\mathrm{-9})$

In the following exercises, evaluate.

$$-x,\text{when}$$

1. ⓐ $x=32$
2. ⓑ $x=\mathrm{-32}$

1. ⓐ −32
2. ⓑ 32

$$-n,\text{when}$$

1. ⓐ $n=20$
2. ⓐ $n=\mathrm{-20}$

Simplify Absolute Values

In the following exercises, simplify.

$$\left|\mathrm{-21}\right|$$

21

$$\left|\mathrm{-42}\right|$$

$$\left|36\right|$$

36

$$-\left|15\right|$$

$$\left|0\right|$$

0

$$-\left|\mathrm{-75}\right|$$

In the following exercises, evaluate.

$$\left|x\right|\text{when}x=\mathrm{-14}$$

14

$$-\left|r\right|\text{when}r=27$$

$$-\left|-y\right|\text{when}y=33$$

−33

$$\left|\mathit{\text{−n}}\right|\text{when}n=\mathrm{-4}$$

In the following exercises, fill in for each of the following pairs of numbers.

$$-\left|\mathrm{-4}\right|\text{\_\_}4$$

<

$$\mathrm{-2}\text{\_\_}\left|\mathrm{-2}\right|$$

$$-\left|\mathrm{-6}\right|\text{\_\_}\mathrm{-6}$$

=

$$-\left|\mathrm{-9}\right|\text{\_\_}\left|\mathrm{-9}\right|$$

In the following exercises, simplify.

$$-(\mathrm{-55})\text{and}-\left|\mathrm{-55}\right|$$

55; −55

$$-(\mathrm{-48})\text{and}-\left|\mathrm{-48}\right|$$

$$\left|12-5\right|$$

7

$$\left|9+7\right|$$

$$6\left|\mathrm{-9}\right|$$

54

$$\left|14\mathrm{-8}\right|-\left|\mathrm{-2}\right|$$

$$\left|9-3\right|-\left|5-12\right|$$

−1

$$5+4\left|15-3\right|$$

Translate Phrases to Expressions with Integers

In the following exercises, translate each of the following phrases into expressions with positive or negative numbers.

the opposite of $16$

−16

the opposite of $\mathrm{-8}$

negative $3$

−3

$19$ minus negative $12$

a temperature of $10$ below zero

−10°

an elevation of $\text{85 feet}$ below sea level

Subtract Integers

Model Subtraction of Integers

In the following exercises, model the following.

$$6-1$$

5

$$\mathrm{-4}-(\mathrm{-3})$$

$$2-(\mathrm{-5})$$

7

$$\mathrm{-1}-4$$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$$24-16$$

8

$$19-(\mathrm{-9})$$

$$\mathrm{-31}-7$$

−38

$$\mathrm{-40}-(\mathrm{-11})$$

$$\mathrm{-52}-(\mathrm{-17})-23$$

−58

$$25-(\mathrm{-3}-9)$$

$$(1-7)-(3-8)$$

−1

$$3^2-7^2$$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

$$x-7\text{when}$$

1. ⓐ $x=5$
2. ⓑ $x=\mathrm{-4}$

1. ⓐ −2
2. ⓑ −11

$$10-y\text{when}$$

1. ⓐ $y=15$
2. ⓑ $y=\mathrm{-16}$

$$2n^2-n+5\text{when}n=\mathrm{-4}$$

41

$$\mathrm{-15}-3u^2\text{when}u=\mathrm{-5}$$

Translate Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

the difference of $\mathrm{-12}\text{and}5$

−12 − 5 = −17

subtract $23$ from $\mathrm{-50}$

Subtract Integers in Applications

In the following exercises, solve the given applications.

Temperature One morning the temperature in Bangor, Maine was $\text{18 degrees.}$ By afternoon, it had dropped $\text{20 degrees.}$ What was the afternoon temperature?

−2 degrees

Temperature On January 4, the high temperature in Laredo, Texas was $\text{78 degrees,}$ and the high in Houlton, Maine was $\mathrm{-28}\text{degrees}.$ What was the difference in temperature of Laredo and Houlton?

Solve Equations using Integers; The Division Property of Equality

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

$$5x-10=\mathrm{-35}$$

1. ⓐ $x=\mathrm{-9}$
2. ⓑ $x=\mathrm{-5}$
3. ⓒ $x=5$

1. ⓐ no
2. ⓑ yes
3. ⓒ no

$$8u+24=\mathrm{-32}$$

1. ⓐ $u=\mathrm{-7}$
2. ⓑ $u=\mathrm{-1}$
3. ⓒ $u=7$

Using the Addition and Subtraction Properties of Equality

In the following exercises, solve.

$$a+14=2$$

−12

$$b-9=\mathrm{-15}$$

$$c+(\mathrm{-10})=\mathrm{-17}$$

−7

$$d-(\mathrm{-6})=\mathrm{-26}$$

Model the Division Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters. Then solve it.

3x = 9; x = 3

Solve Equations Using the Division Property of Equality

In the following exercises, solve each equation using the division property of equality and check the solution.

$$8p=72$$

9

$$\mathrm{-12}q=48$$

$$\mathrm{-16}r=\mathrm{-64}$$

4

$$\mathrm{-5}s=\mathrm{-100}$$

Translate to an Equation and Solve.

In the following exercises, translate and solve.

$$\text{The product of −6 and}y\text{is}\mathrm{-42}$$

−6y = −42; y = 7

$$\text{The difference of}z\text{and −13 is −18.}$$

Four more than $m$ is $\mathrm{-48}.$

m + 4 = −48; m = −52

$$\text{The product of −21 and}n\text{is 63.}$$

Chapter Practice Test

In the following exercises, compare the numbers, using $<\text{or}>\text{or}\text{=.}$

In the following exercises, find the opposite of each number.

In the following exercises, simplify.

In the following exercises, evaluate.

In the following exercises, translate each phrase into an algebraic expression and then simplify, if possible.

In the following exercises, solve.

In the following exercises, solve.

In the following exercises, translate and solve.