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.

Determine whether each of the following is a solution of

2 x - 5 = −13 :

ⓐ $x = 4$
ⓑ $x = −4$
ⓒ $x = −9 .$

Solution

ⓐ Substitute 4 for x in the equation to determine if it is true.


Multiply.
Subtract.

Since $x = 4$ does not result in a true equation, $4$ is not a solution to the equation.

ⓑ Substitute −4 for x in the equation to determine if it is true.

Multiply.
Subtract.

Since $x = −4$ results in a true equation, $−4$ is a solution to the equation.

ⓒ Substitute −9 for x in the equation to determine if it is true.

Substitute −9 for x.
Multiply.
Subtract.

Since $x = −9$ does not result in a true equation, $−9$ is not a solution to the equation.

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.

This figure has two columns. The first column has the equation x plus 4 equals 12. Underneath there is x plus 4 minus 4 equals 12 minus 4. Under this there is x equals 8. The second column has the equation y minus 5 equals 9. Underneath there is the equation y minus 5 plus 5 equals 9 plus 5. Under this there is y equals 14.

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

Solve: $y + 9 = 5 .$

Solution


Subtract 9 from each side to undo the addition.
Simplify.

Check the result by substituting $−4$ into the original equation.

Demonstration of substituting a value into an algebraic equation to verify its solution.
	$y + 9 = 5$
Substitute −4 for y	$−4 + 9 \overset{?}{=} 5$
	$5 = 5 ✓$

Since $y = −4$ makes $y + 9 = 5$ a true statement, we found the solution to this equation.

Solve: $a - 6 = −8$

Solution


Add 6 to each side to undo the subtraction.
Simplify.
Check the result by substituting $−2$ into the original equation:
Substitute $−2$ for $a$

The solution to $a - 6 = −8$ is $−2 .$

Since $a = −2$ makes $a - 6 = −8$ a true statement, we found the solution to this equation.

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 1.

This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed.

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 $6 \div 2 = 3) .$

What equation models the situation shown in Figure 2? 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 $2 x = 6 .$

This image has two columns. In the first column are two identical envelopes. In the second column there are six blue circles, randomly placed. Under the figure is two times x equals 6.

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

This figure has two rows. The first row has the equation 2x divided by 2 equals 6 divided by 2. The second row has the equation x equals 3.

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.

Figure 3 shows another example.

This image has two columns. In the first column are three envelopes. In the second column there are four rows of three blue circles. Underneath the image is the equation 3x equals 12.

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

This image has two columns. In the first column are four envelopes. In the second column there are twelve blue circles.

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

This image shows the equation 3x divided by 3 equals 12 divided by 3. Below this equation is the equation x equals 4.

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

Write an equation modeled by the envelopes and counters, and then solve it.

This image has two columns. In the first column are four envelopes. In the second column there are 8 blue circles.

Solution

There are

4 envelopes,

or

4

unknown values, on the left that match the

8 counters

on the right. Let’s call the unknown quantity in the envelopes

x .

Write the equation.
Divide both sides by 4.
Simplify.

There are $2 counters$ in each envelope.

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.

$Solve: 7 x = −49 .$

Solution

To isolate

x,

we need to undo multiplication.


Divide each side by 7.
Simplify.

Check the solution.

Verification of a linear equation solution through substitution.
	$7 x = −49$
Substitute −7 for x.	$7 (−7) \overset{?}{=} −49$
	$−49 = −49 ✓$

Therefore, $−7$ is the solution to the equation.

Solve: $−3 y = 63 .$

Solution

To isolate

y,

we need to undo the multiplication.


Divide each side by −3.
Simplify

Check the solution.

Demonstrates checking the solution y = -21 for the equation -3y = 63, verifying the equality of both sides.
	$−3 y = 63$
Substitute −21 for y.	$−3 (−21) \overset{?}{=} 63$
	$63 = 63 ✓$

Since this is a true statement, $y = −21$ is the solution to the equation.

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.

Translate and solve: five more than $x$ is equal to $−3 .$

Solution

This table illustrates the step-by-step process of translating a verbal statement into a linear equation and solving it.
	five more than $x$ is equal to $−3$
Translate	$x + 5 = −3$
Subtract $5$ from both sides.	$x + 5 - 5 = −3 - 5$
Simplify.	$x = −8$

Check the answer by substituting it into the original equation.

$\begin{array}{l} x + 5 = −3 \\ −8 + 5 \overset{?}{=} −3 \\ −3 = −3 ✓ \end{array}$

Translate and solve: the difference of $n$ and $6$ is $−10 .$

Solution

This table illustrates the step-by-step process of translating a verbal problem into a linear equation and solving for the variable 'n'.
	the difference of $n$ and $6$ is $−10$
Translate.	$n - 6 = −10$
Add $6$ to each side.	$n - 6 + 6 = −10 + 6$
Simplify.	$n = −4$

Check the answer by substituting it into the original equation.

$\begin{matrix} n - 6 = −10 \\ −4 - 6 \overset{?}{=} −10 \\ −10 = −10 ✓ \end{matrix}$

Translate and solve: the number $108$ is the product of $−9$ and $y .$

Solution

Detailed steps to solve the linear equation 108 = -9y, starting from its word problem translation.
	the number of $108$ is the product of $−9$ and $y$
Translate.	$108 = −9 y$
Divide by $−9$ .	$\frac{108}{−9} = \frac{−9 y}{−9}$
Simplify.	$−12 = y$

Check the answer by substituting it into the original equation.

$\begin{matrix} 108 = −9 y \\ 108 \overset{?}{=} −9 (−12) \\ 108 = 108 ✓ \end{matrix}$

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

$For any numbers a, b, c,$
$if a = b then a - c = b - c .$ $For any numbers a, b, c,$
$if a = b then a + c = b + c .$
Division Property of Equality
- For any numbers $a, b, c,$ and $c \neq 0$
  If $a = b$ , then $\frac{a}{c} = \frac{b}{c}$ .

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

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.

4 x - 2 = 6

ⓐ $x = −2$
ⓑ $x = −1$
ⓒ $x = 2$

Solution

ⓐ no
ⓑ no
ⓒ yes

4 y - 10 = −14

ⓐ $y = −6$
ⓑ $y = −1$
ⓒ $y = 1$

9 a + 27 = −63

ⓐ $a = 6$
ⓑ $a = −6$
ⓒ $a = −10$

Solution

ⓐ no
ⓑ no
ⓒ yes

7 c + 42 = −56

ⓐ $c = 2$
ⓑ $c = −2$
ⓒ $c = −14$

Solve Equations Using the Addition and Subtraction Properties of Equality

In the following exercises, solve for the unknown.

$n + 12 = 5$

Solution

n = −7

$m + 16 = 2$

$p + 9 = −8$

Solution

p = −17

$q + 5 = −6$

$u - 3 = −7$

Solution

u = −4

$v - 7 = −8$

$h - 10 = −4$

Solution

h = 6

$k - 9 = −5$

$x + (−2) = −18$

Solution

x = −16

$y + (−3) = −10$

$r - (−5) = −9$

Solution

r = −14

$s - (−2) = −11$

Model the Division Property of Equality

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

Three light blue envelopes are stacked vertically on the left side of a white background, while six light yellow circular tokens are arranged in two columns of three on the right.

Solution

3x = 6; x = 2

Two light blue envelopes are paired with ten light yellow circles arranged in two columns of five on a white background.

Two light blue envelopes are paired with eight light yellow circles arranged in two columns of four on a white background.

Solution

2x = 8; x = 4

Three light blue envelopes are displayed on the left, while nine yellow-rimmed circles are arranged in a 3x3 grid on the right, all on a white background.

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.

$5 x = 45$

Solution

x = 9

$4 p = 64$

$−7 c = 56$

Solution

c = −8

$−9 x = 54$

$−14 p = −42$

Solution

p = 3

$−8 m = −40$

$−120 = 10 q$

Solution

q = −12

$−75 = 15 y$

$24 x = 480$

Solution

x = 20

$18 n = 540$

$−3 z = 0$

Solution

z = 0

$4 u = 0$

Translate to an Equation and Solve

In the following exercises, translate and solve.

Four more than $n$ is equal to 1.

Solution

n + 4 = 1; n = −3

Nine more than $m$ is equal to 5.

The sum of eight and $p$ is $−3$ .

Solution

8 + p = −3; p = −11

The sum of two and $q$ is $−7$ .

The difference of $a$ and three is $−14$ .

Solution

a − 3 = −14; a = −11

The difference of $b$ and $5$ is $−2$ .

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

Solution

−42 = −7x; x = 6

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

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

Solution

−15f = 75; f = −5

The product of −18 and $g$ is 36.

−6 plus $c$ is equal to 4.

Solution

−6 + c = 4; c = 10

−2 plus $d$ is equal to 1.

Nine less than $m$ is −4.

Solution

m − 9 = −4; m = 5

Thirteen less than $n$ is $−10$ .

Mixed Practice

In the following exercises, solve.

ⓐ $x + 2 = 10$
ⓑ $2 x = 10$

Solution

ⓐ x = 8
ⓑ x = 5

ⓐ $y + 6 = 12$
ⓑ $6 y = 12$

ⓐ $−3 p = 27$
ⓑ $p - 3 = 27$

Solution

ⓐ p = −9
ⓑ p = 30

ⓐ $−2 q = 34$
ⓑ $q - 2 = 34$

$a - 4 = 16$

Solution

a = 20

$b - 1 = 11$

$−8 m = −56$

Solution

m = 7

$−6 n = −48$

$−39 = u + 13$

Solution

u = −52

$−100 = v + 25$

$11 r = −99$

Solution

r = −9

$15 s = −300$

$100 = 20 d$

Solution

d = 5

$250 = 25 n$

$−49 = x - 7$

Solution

x = −42

$64 = y - 4$

Everyday Math

Cookie packaging A package of $51 cookies$ has $3$ equal rows of cookies. Find the number of cookies in each row, $c,$ by solving the equation $3 c = 51 .$

Solution

17 cookies

Kindergarten class Connie’s kindergarten class has $24 children.$ She wants them to get into $4$ equal groups. Find the number of children in each group, $g,$ by solving the equation $4 g = 24 .$

Writing Exercises

Is modeling the Division Property of Equality with envelopes and counters helpful to understanding how to solve the equation $3 x = 15 ?$ Explain why or why not.

Solution

Sample answer: It is helpful because it shows how the counters can be divided among the envelopes.

Suppose you are using envelopes and counters to model solving the equations $x + 4 = 12$ and $4 x = 12 .$ Explain how you would solve each equation.

Frida started to solve the equation $−3 x = 36$ by adding $3$ to both sides. Explain why Frida’s method will not solve the equation.

Solution

Sample answer: The operation used in the equation is multiplication. The inverse of multiplication is division, not addition.

Raoul started to solve the equation $4 y = 40$ by subtracting $4$ from both sides. Explain why Raoul’s method will not solve the equation.

Self Check

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

A self-assessment checklist for math skills, asking users to rate their ability to solve equations with integers and understand properties of equality using options like 'Confidently', 'With some help', or 'No-I don't get it!'.

ⓑ 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.

$5$

Solution

This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at 5.

$−5$

$−3$

Solution

This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at negative 3.

$3$

$−8$

Solution

This figure is a number line. It is scaled from negative 10 to 10 in increments of 2. There is a point at negative 8.

$−7$

Order Positive and Negative Numbers

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

$4__8$

Solution

<

$−6__3$

$−5__−10$

Solution

>

$−9__−4$

$2__−7$

Solution

>

$−3__1$

Find Opposites

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

$6$

Solution

−6

$−2$

$−4$

Solution

4

$3$

In the following exercises, simplify.

ⓐ $- (8)$
ⓑ $- (−8)$

Solution

ⓐ −8
ⓑ 8

ⓐ $- (9)$
ⓑ $- (−9)$

In the following exercises, evaluate.

- x, when

ⓐ $x = 32$
ⓑ $x = −32$

Solution

ⓐ −32
ⓑ 32

- n, when

ⓐ $n = 20$
ⓐ $n = −20$

Simplify Absolute Values

In the following exercises, simplify.

$| −21 |$

Solution

21

$| −42 |$

$| 36 |$

Solution

36

$- | 15 |$

$| 0 |$

Solution

0

$- | −75 |$

In the following exercises, evaluate.

$| x | when x = −14$

Solution

14

$- | r | when r = 27$

$- | - y | when y = 33$

Solution

−33

$| −n | when n = −4$

In the following exercises, fill in $<, >, or =$ for each of the following pairs of numbers.

$- | −4 |__4$

Solution

<

$−2__| −2 |$

$- | −6 |__−6$

Solution

=

$- | −9 |__| −9 |$

In the following exercises, simplify.

$- (−55) and - | −55 |$

Solution

55; −55

$- (−48) and - | −48 |$

$| 12 - 5 |$

Solution

7

$| 9 + 7 |$

$6 | −9 |$

Solution

54

$| 14 −8 | - | −2 |$

$| 9 - 3 | - | 5 - 12 |$

Solution

−1

$5 + 4 | 15 - 3 |$

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 $16$

Solution

−16

the opposite of $−8$

negative $3$

Solution

−3

$19$ minus negative $12$

a temperature of $10$ below zero

Solution

−10°

an elevation of $85 feet$ below sea level

Add Integers

Model Addition of Integers

In the following exercises, model the following to find the sum.

$3 + 7$

Solution

10

$−2 + 6$

$5 + (−4)$

Solution

1

$−3 + (−6)$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$14 + 82$

Solution

96

$−33 + (−67)$

$−75 + 25$

Solution

−50

$54 + (−28)$

$11 + (−15) + 3$

Solution

−1

$−19 + (−42) + 12$

$−3 + 6 (−1 + 5)$

Solution

21

$10 + 4 (−3 + 7)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

n + 4 when

ⓐ $n = −1$
ⓑ $n = −20$

Solution

ⓐ 3
ⓑ −16

x + (−9) when

ⓐ $x = 3$
ⓑ $x = −3$

${(x + y)}^{3} when x = −4, y = 1$

Solution

−27

${(u + v)}^{2} when u = −4, v = 11$

Translate Word Phrases to Algebraic Expressions

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

$the sum of −8 and 2$

Solution

−8 + 2 = −6

$4 more than −12$

$10 more than the sum of −5 and −6$

Solution

10 + [−5 + (−6)] = −1

$the sum of 3 and −5, increased by 18$

Add Integers in Applications

In the following exercises, solve.

Temperature On Monday, the high temperature in Denver was $−4 degrees.$ Tuesday’s high temperature was $20 degrees$ more. What was the high temperature on Tuesday?

Solution

16 degrees

Credit Frida owed $$75$ on her credit card. Then she charged $$21$ more. What was her new balance?

Subtract Integers

Model Subtraction of Integers

In the following exercises, model the following.

$6 - 1$

Solution

This figure is a row of 6 light pink circles, representing positive counters. The first one is circled.
5

$−4 - (−3)$

$2 - (−5)$

Solution

This figure shows 2 rows. The first row shows 7 light pink circles, representing positive counters. The second row shows 5 dark pink circles, representing negative counters. The entire second row is circled.
7

$−1 - 4$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$24 - 16$

Solution

8

$19 - (−9)$

$−31 - 7$

Solution

−38

$−40 - (−11)$

$−52 - (−17) - 23$

Solution

−58

$25 - (−3 - 9)$

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

Solution

−1

$3^{2} - 7^{2}$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

x - 7 when

ⓐ $x = 5$
ⓑ $x = −4$

Solution

ⓐ −2
ⓑ −11

10 - y when

ⓐ $y = 15$
ⓑ $y = −16$

$2 n^{2} - n + 5 when n = −4$

Solution

41

$−15 - 3 u^{2} when u = −5$

Translate Phrases to Algebraic Expressions

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

the difference of $−12 and 5$

Solution

−12 − 5 = −17

subtract $23$ from $−50$

Subtract Integers in Applications

In the following exercises, solve the given applications.

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

Solution

−2 degrees

Temperature On January 4, the high temperature in Laredo, Texas was $78 degrees,$ and the high in Houlton, Maine was $−28 degrees .$ What was the difference in temperature of Laredo and Houlton?

Multiply and Divide Integers

Multiply Integers

In the following exercises, multiply.

$−9 \cdot 4$

Solution

−36

$5 (−7)$

$(−11) (−11)$

Solution

121

$−1 \cdot 6$

Divide Integers

In the following exercises, divide.

$56 \div (−8)$

Solution

−7

$−120 \div (−6)$

$−96 \div 12$

Solution

−8

$96 \div (−16)$

$45 \div (−1)$

Solution

−45

$−162 \div (−1)$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$5 (−9) - 3 (−12)$

Solution

−9

${(−2)}^{5}$

$- 3^{4}$

Solution

−81

$(−3) (4) (−5) (−6)$

$42 - 4 (6 - 9)$

Solution

54

$(8 - 15) (9 - 3)$

$−2 (−18) \div 9$

Solution

4

$45 \div (−3) - 12$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

$7 x - 3 when x = −9$

Solution

−66

$16 - 2 n when n = −8$

$5 a + 8 b when a = −2, b = −6$

Solution

−58

$x^{2} + 5 x + 4 when x = −3$

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate to an algebraic expression and simplify if possible.

the product of $−12$ and $6$

Solution

−12(6) = −72

the quotient of $3$ and the sum of $−7$ and $s$

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.

5 x - 10 = −35

ⓐ $x = −9$
ⓑ $x = −5$
ⓒ $x = 5$

Solution

ⓐ no
ⓑ yes
ⓒ no

8 u + 24 = −32

ⓐ $u = −7$
ⓑ $u = −1$
ⓒ $u = 7$

Using the Addition and Subtraction Properties of Equality

In the following exercises, solve.

$a + 14 = 2$

Solution

−12

$b - 9 = −15$

$c + (−10) = −17$

Solution

−7

$d - (−6) = −26$

Model the Division Property of Equality

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

This image has two columns. In the first column there are three envelopes. In the second column there are two vertical rows. The first row includes five blue circles, the second row includes four blue circles.

Solution

3x = 9; x = 3

This figure has two columns. In the first column there are two envelopes. In the second column there are two vertical rows, each includes four blue circles.

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.

$8 p = 72$

Solution

9

$−12 q = 48$

$−16 r = −64$

Solution

4

$−5 s = −100$

Translate to an Equation and Solve.

In the following exercises, translate and solve.

$The product of −6 and y is −42$

Solution

−6y = −42; y = 7

$The difference of z and −13 is −18.$

Four more than $m$ is $−48 .$

Solution

m + 4 = −48; m = −52

$The product of −21 and n is 63.$

Everyday Math

Describe how you have used two topics from this chapter in your life outside of your math class during the past month.

Solution

Answers will vary.

Chapter Practice Test

Locate and label $0, 2, −4,$ and $−1$ on a number line.

In the following exercises, compare the numbers, using $< or > or =.$

ⓐ $−6__3$
ⓑ $−1__−4$

Solution

ⓐ <
ⓑ >

ⓐ $−5__| −5 |$
ⓑ $- | −2 |__−2$

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

ⓐ $−7$
ⓑ $8$

Solution

ⓐ 7
ⓑ −8

In the following exercises, simplify.

$- (−22)$

$| 4 - 9 |$

Solution

5

$−8 + 6$

$−15 + (−12)$

Solution

−27

$−7 - (−3)$

$10 - (5 - 6)$

Solution

11

$−3 \cdot 8$

$−6 (−9)$

Solution

54

$70 \div (−7)$

${(−2)}^{3}$

Solution

−8

$- 4^{2}$

$16 −3 (5 −7)$

Solution

22

$| 21 - 6 | - | −8 |$

In the following exercises, evaluate.

$35 - a when a = −4$

Solution

39

${(−2 r)}^{2} when r = 3$

$3 m - 2 n when m = 6, n = −8$

Solution

34

$- | - y | when y = 17$

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

the difference of −7 and −4

Solution

−7 − (−4) = −3

the quotient of $25$ and the sum of $m$ and $n .$

In the following exercises, solve.

Early one morning, the temperature in Syracuse was $−8 °F.$ By noon, it had risen $12°.$ What was the temperature at noon?

Solution

4°F

Collette owed $$128$ on her credit card. Then she charged $$65.$ What was her new balance?

In the following exercises, solve.

$n + 6 = 5$

Solution

n = −1

$p - 11 = −4$

$−9 r = −54$

Solution

r = 6

In the following exercises, translate and solve.

$The product of 15 and x is 75.$

$Eight less than y is −32.$

Solution

y − 8 = −32; y = −24