# Spectrum Math Grade 8 Chapter 3 Lesson 4 Answer Key Solving Complex 1-Variable Equations

Students can use the Spectrum Math Grade 8 Answer Key Chapter 3 Lesson 3.4 Solving Complex 1-Variable Equations as a quick guide to resolve any of their doubts.

## Spectrum Math Grade 8 Chapter 3 Lesson 3.4 Solving Complex 1-Variable Equations Answers Key

Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
2n – 7 = 19 5 + 5 = 11
First, undo the subtraction by adding.
2n – 7 + 7 = 19 + 7 2n = 26
First, undo the addition by subtracting.
$$\frac{n}{3}$$ + 5 – 5 = 11 – 5 $$\frac{n}{3}$$ = 6
Then, undo the multiplication by dividing.
n = 13
Then, undo the division by multiplying.
$$\frac{n}{3}$$ × 3 = 6 × 3 n = 18

Find the value of the variable in each equation.

Question 1.
a. 2n + 2 = 16 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
2n + 2 = 16
First, undo the addition by subtracting.
2n + 2 – 2 = 16 – 2
2n = 14
Then, undo the multiplication by dividing.
2n ÷ 2 = 14 ÷ 2
n = 7

b. $$\frac{a}{3}$$ – 1 = 4 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{a}{3}$$ – 1 = 4
First, undo the subtraction by adding.
$$\frac{a}{3}$$ – 1 + 1 = 4 + 1
$$\frac{a}{3}$$  = 5
Then, undo the division by multiplying.
$$\frac{a}{3}$$ x 3  = 5 x 3
a = 15

c. $$\frac{b}{4}$$ + 2 = 11 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{b}{4}$$ + 2 = 11
First, undo the addition by subtracting.
$$\frac{b}{4}$$ + 2 – 2 = 11 – 2
$$\frac{b}{4}$$  = 9
Then, undo the division by multiplying.
$$\frac{b}{4}$$  x 4 = 9 x 4
b = 36

Question 2.
a. 11p – 5 = 28 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
11p – 5 = 28
First, undo the subtraction by adding.
11p – 5 + 5 = 28 + 5
11p = 33
Then, undo the multiplication by dividing.
$$\frac{11 p}{11}$$   =  $$\frac{33}{11}$$
p = 3

b. 8b + 12 = 52 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
8b + 12 = 52
First, undo the addition by subtracting.
8b + 12 – 12 = 52 – 12
8b = 40
Then, undo the multiplication by dividing.
8b ÷ 8 = 40 ÷ 8
b = 5

c. $$\frac{r}{20}$$ – 3 = 3 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{r}{20}$$ – 3 = 3
First, undo the subtraction by adding.
$$\frac{r}{20}$$ – 3 + 3= 3 + 3
$$\frac{r}{20}$$  = 6
Then, undo the division by multiplying.
$$\frac{r}{20}$$  x 20 = 6 x 20
r = 120

Question 3.
a. $$\frac{m}{16}$$ + 7 = 10 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{m}{16}$$ + 7 = 10
First, undo the addition by subtracting.
$$\frac{m}{16}$$ + 7 – 7 = 10 – 7
$$\frac{m}{16}$$  = 3
Then, undo the division by multiplying.
$$\frac{m}{16}$$  x 16 = 3 x 16
m = 48

b. 6n + 4 = 64 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
6n + 4 = 64
First, undo the addition by subtracting.
6n + 4  – 4= 64 – 4
6n = 60
Then, undo the multiplication by dividing.
6n ÷ 6 = 60 ÷ 6
n = 10

c. 4s – 5 = 39 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
4s – 5 = 39
First, undo the subtraction by adding.
4s – 5 + 5 = 39 + 5
4s = 44
Then, undo the multiplication by dividing.
4s ÷ 4 = 44 ÷ 4
s = 11

Question 4.
a. $$\frac{a}{9}$$ – 3 = 6 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{a}{9}$$ – 3 = 6
First, undo the subtraction by adding.
$$\frac{a}{9}$$ – 3 + 3= 6 + 3
$$\frac{a}{9}$$  = 9
Then, undo the division by multiplying.
$$\frac{a}{9}$$  x 9 = 9 x 9
a = 81

b. 5d + 6 = 71 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
5d + 6 = 71
First, undo the addition by subtracting.
5d + 6 – 6 = 71 – 6
5d = 65
Then, undo the multiplication by dividing.
5d ÷ 5 = 65 ÷ 5
d = 13

c. $$\frac{m}{16}$$ + 5 = 14 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{m}{16}$$ + 5 = 14
First, undo the addition by subtracting.
$$\frac{m}{16}$$ + 5 – 5= 14 – 5
$$\frac{m}{16}$$  = 9
Then, undo the division by multiplying.
$$\frac{m}{16}$$  x 16 = 9 x 16
m = 144

Question 5.
a. 9a – 11 = 61 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
9a – 11 = 61
First, undo the subtraction by adding.
9a – 11 + 11 = 61 + 11
9a = 72
Then, undo the multiplication by dividing.
9a ÷ 9 = 72 ÷ 9
a = 8

b. $$\frac{e}{12}$$ – 7 = 3
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{e}{12}$$ – 7 = 3
First, undo the subtraction by adding.
$$\frac{e}{12}$$ – 7  + 7 = 3 + 7
$$\frac{e}{12}$$  = 10
Then, undo the division by multiplying.
$$\frac{e}{12}$$  x 12= 10 x 12
e = 120

c. $$\frac{i}{4}$$ + 5 = 73 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{i}{4}$$ + 5 = 73
First, undo the addition by subtracting.
$$\frac{i}{4}$$ + 5 – 5 = 73 – 5
$$\frac{i}{4}$$ = 68
Then, undo the division by multiplying.
$$\frac{i}{4}$$ x 4 = 68 x 4
i = 272

Question 6.
a. 3p + 12 = 54 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
3p + 12 = 54
First, undo the addition by subtracting.
3p + 12 – 12 = 54 – 12
3p = 42
Then, undo the multiplication by dividing.
3p ÷ 3 = 42 ÷ 3
p = 14

b. $$\frac{n}{3}$$ + 12 = 27 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{n}{3}$$ + 12 = 27
First, undo the addition by subtracting.
$$\frac{n}{3}$$ + 12 – 12 = 27 -12
$$\frac{n}{3}$$ = 15
Then, undo the division by multiplying.
$$\frac{n}{3}$$ x3 = 15 x 3
n = 45

c. 5b – 7 = 93 _____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
5b – 7 = 93
First, undo the subtraction by adding.
5b – 7 + 7  = 93 + 7
5b = 100
Then, undo the multiplication by dividing.
5b ÷ 5 = 100 ÷ 5
b = 20

Question 7.
a. $$\frac{s}{15}$$ + 1 = 5 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{s}{15}$$ + 1 = 5
First, undo the addition by subtracting.
$$\frac{s}{15}$$ + 1 – 1 = 5 – 1
$$\frac{s}{15}$$  = 4
Then, undo the division by multiplying.
$$\frac{s}{15}$$ x 15 = 15 x 4
s = 60

b. 6x + 25 = 73 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
6x + 25 = 73
First, undo the addition by subtracting.
6x + 25 – 25 = 73 – 25
6x = 48
Then, undo the multiplication by dividing.
6x ÷ 6 = 48 ÷ 6
x = 8

c. $$\frac{a}{3}$$ – 3 = 11 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{a}{3}$$ – 3 = 11
First, undo the subtraction by adding.
$$\frac{a}{3}$$ – 3 + 3 = 11 + 3
$$\frac{a}{3}$$  = 13
Then, undo the division by multiplying.
$$\frac{a}{3}$$  x 3 = 13 x 3
a = 39

Question 8.
a. 3r – 11 = 43 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
3r – 11 = 43
First, undo the subtraction by adding.
3r – 11 + 11 = 43 +11
3r = 54
Then, undo the multiplication by dividing.
3r ÷ 3 = 54 ÷ 3
r = 18

b. $$\frac{x}{7}$$ + 14 = 22 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{x}{7}$$ + 14 = 22
First, undo the addition by subtracting.
$$\frac{x}{7}$$ + 14 – 14= 22 -14
$$\frac{x}{7}$$ = 8
Then, undo the division by multiplying.
$$\frac{x}{7}$$ x 7 = 7 x 8
x = 56

c. 5m + 13 = 68 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
5m + 13 = 68
First, undo the addition by subtracting.
5m + 13 – 13  = 68 – 13
5m = 55
Then, undo the multiplication by dividing.
5m ÷ 5 = 55 ÷ 5
m = 11

Question 9.
a. $$\frac{n}{5}$$ – 5 = 8 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{n}{5}$$ – 5 = 8
First, undo the subtraction by adding.
$$\frac{n}{5}$$ – 5 + 5 = 8 + 5
$$\frac{n}{5}$$  = 14
Then, undo the division by multiplying.
$$\frac{n}{5}$$  x 5 = 14 x 5
n = 70

b. $$\frac{a}{6}$$ + 4 = 20 _____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{a}{6}$$ + 4 = 20
First, undo the addition by subtracting.
$$\frac{a}{6}$$ + 4 – 4 = 20 – 4
$$\frac{a}{6}$$ = 16
Then, undo the division by multiplying.
$$\frac{a}{6}$$ x 6 = 16 x 6
a = 96

c. 3p – 15 = 48 _____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
3p – 15 = 48
First, undo the subtraction by adding.
3p – 15 + 15 = 48 + 15
3p = 36
Then, undo the multiplication by dividing.
3p ÷ 3 = 36 ÷ 3
p = 12

Question 10.
a. $$\frac{n}{5}$$ – 5 = 8 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{n}{5}$$ – 5 = 8
First, undo the subtraction by adding.
$$\frac{n}{5}$$ – 5 + 5 = 8 + 5
$$\frac{n}{5}$$  = 14
Then, undo the division by multiplying.
$$\frac{n}{5}$$  x 5 = 14 x 5
n = 70

b. $$\frac{a}{6}$$ + 4 = 20 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{a}{6}$$ + 4 = 20
First, undo the addition by subtracting.
$$\frac{a}{6}$$ + 4 – 4 = 20 – 4
$$\frac{a}{6}$$ = 16
Then, undo the division by multiplying.
$$\frac{a}{6}$$ x 6 = 16 x 6
a = 96

c. 3p – 15 = 48 _______
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
3p – 15 = 48
First, undo the subtraction by adding.
3p – 15 + 15 = 48 + 15
3p = 36
Then, undo the multiplication by dividing.
3p ÷ 3 = 36 ÷ 3
p = 12

Find the value of the variable in each equation.

Question 1.
a. $$\frac{a}{10}$$ + 4 = 5 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{a}{10}$$ + 4 = 5
First, undo the addition by subtracting.
$$\frac{a}{10}$$ + 4 – 4 = 5 – 4
$$\frac{a}{10}$$ = 1
Then, undo the division by multiplying.
$$\frac{a}{10}$$ x 10 = 1 x 10
a = 100

b. $$\frac{c}{2}$$ + 5 = 3 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{c}{2}$$ + 5 = 3
First, undo the addition by subtracting.
$$\frac{c}{2}$$ + 5  – 5 = 3 – 5
$$\frac{c}{2}$$ = -2
Then, undo the division by multiplying.
$$\frac{c}{2}$$ x 2 = -2 x 2
c = -4

c. 3e – 2 = -29 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
3e – 2 = -29
First, undo the subtraction by adding.
3e – 2 + 2 = -29 + 2
3e = -27
Then, undo the multiplication by dividing.
3e ÷ 3 = -27 ÷ 3
e = -9

Question 2.
a. 1 – g = -5 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
1 – g = -5
First, undo the addition by subtracting.
1 – g  – 1 = -5 – 1
-g = -6
Then, undo the multiplication by dividing.
-g ÷ -1 = -6 ÷ -1
g = 6

b. $$\frac{h-10}{2}$$ = -7 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{h-10}{2}$$ = -7
First, undo the division by multiplying.
$$\frac{h-10}{2}$$  x 2 = -7 x 2
h-10 = -14
Then, undo the subtraction by adding.
h – 10 + 10 = -14 + 10
h = -4

c. $$\frac{j-5}{2}$$ = 5 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{j-5}{2}$$ = 5
First, undo the division by multiplying.
$$\frac{j-5}{2}$$ x 2 = 5 x 2
j – 5 = 10
Then, undo the subtraction by adding.
j – 5 + 5 = 10 + 5
j = 15

Question 3.
a. -9 + $$\frac{f}{4}$$ = -7 _______
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
-9 + $$\frac{f}{4}$$ = -7
First, undo the subtraction by adding.
-9 + $$\frac{f}{4}$$ + 9 = -7 + 9
$$\frac{f}{4}$$  = 2
Then, undo the division by multiplying.
$$\frac{f}{4}$$  x 4 = 2  x 4
f = 8

b. $$\frac{9+n}{3}$$ = 2 _______
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{9+n}{3}$$ = 2
First, undo the division by multiplying.
$$\frac{9+n}{3}$$ x 3  = 2 x 3
9 + n = 6
Then, undo the addition by subtracting.
9 + n – 9 = 6 – 9
n = -3

c. $$\frac{-5+p}{22}$$ = -1 _______
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{-5+p}{22}$$ = -1
First, undo the division by multiplying.
$$\frac{-5+p}{22}$$ x 22 = -1 x 22
-5 + p = -22
Then, undo the subtraction by adding.
-5 + p + 5  = -22 + 5
p = -17

Question 4.
a. 4q – 9 = -9 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
4q – 9 = -9
First, undo the subtraction by adding.
4q – 9 + 9 = -9 + 9
4q = 0
Then, undo the multiplication by dividing.
4q ÷ 4 = 0 ÷ 4
q = 0

b. $$\frac{s+9}{2}$$ = 3 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{s+9}{2}$$ = 3
First, undo the division by multiplying.
$$\frac{s+9}{2}$$ x 2 = 3 x 2
s + 9 = 6
Then, undo the addition by subtracting.
s + 9 – 9  = 6 – 9
s = -3

c. $$\frac{-12+u}{11}$$ = -3 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{-12+u}{11}$$ = -3
First, undo the division by multiplying.
$$\frac{-12+u}{11}$$ x 11 = -3 x 11
-12 + u = -33
Then, undo the subtraction by adding.
-12 + u + 12 = -33 + 12
u = -21

Question 5.
a. $$\frac{-4+w}{2}$$ = 6 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{-4+w}{2}$$ = 6
First, undo the division by multiplying.
$$\frac{-4+w}{2}$$ x 2 = 6 x 2
-4 + w = 12
Then, undo the subtraction by adding.
-4 + w + 4 = 12 + 4
w = 16

b. -5 + $$\frac{y}{3}$$ = 0 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
-5 + $$\frac{y}{3}$$ = 0
First, undo the subtraction by adding.
-5 + $$\frac{y}{3}$$  + 5 = 0 + 5
$$\frac{y}{3}$$  = 5
Then, undo the division by multiplying.
$$\frac{y}{3}$$  x 3 = 5  x 3
y = 15

c. $$\frac{b}{4}$$ + 8 = 7 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{b}{4}$$ + 8 = 7
First, undo the addition by subtracting.
$$\frac{b}{4}$$ + 8 – 8 = 7 – 8
$$\frac{b}{4}$$ = -1
Then, undo the division by multiplying.
$$\frac{b}{4}$$ x 4 = -1 x 4
b = -4

Question 6.
a. 9 + $$\frac{d}{4}$$ = 15 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
9 + $$\frac{d}{4}$$ = 15
First, undo the addition by subtracting.
9 + $$\frac{d}{4}$$ – 9 = 15 – 9
$$\frac{d}{4}$$ =6
Then, undo the division by multiplying.
$$\frac{d}{4}$$ x 4 = 6 x 4
d = 24

b. 6 + $$\frac{f}{2}$$ = 15 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
6 + $$\frac{f}{2}$$ = 15
First, undo the addition by subtracting.
6 + $$\frac{f}{2}$$ – 6 = 15 – 6
$$\frac{f}{4}$$ = 9
Then, undo the division by multiplying.
$$\frac{f}{4}$$ x 4 = 9 x 4
f = 36

c. $$\frac{h+11}{3}$$ = -2 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{h+11}{3}$$ = -2
First, undo the division by multiplying.
$$\frac{h+11}{3}$$ x 3 = -2 x 3
h + 11 = -6
Then, undo the addition by subtracting.
h + 11 – 11  = 6 – 11
h = -5

Question 7.
a. $$\frac{j-10}{3}$$ = -4 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{j-10}{3}$$ = -4
First, undo the division by multiplying.
$$\frac{j-10}{3}$$  x 3  = -4 x 3
j-10 = -12
Then, undo the subtraction by adding.
j-10 + 10 = -12 + 10
j = -2

b. -12k + 4 = 100 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
-12k + 4 = 100
First, undo the addition by subtracting.
-12k + 4 – 4  = 100 – 4
-12k = 96
Then, undo the multiplication by dividing.
-12k ÷ -12 = 96 ÷ -12
k = -8

c. $$\frac{m}{16}$$ – 9 = -8 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{m}{16}$$ – 9 = -8
First, undo the subtraction by adding.
$$\frac{m}{16}$$ – 9 + 9 = -8 + 9
$$\frac{m}{16}$$  = 1
Then, undo the division by multiplying.
$$\frac{m}{16}$$  x 16 = 1  x 16
m = 16

Question 8.
a. -7 + 4o = -15 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
-7 + 4o = -15
First, undo the subtraction by adding.
-7 + 4o + 7 = -15 + 7
4o = -8
Then, undo the multiplication by dividing.
4o ÷ 4 = -8 ÷ 4
o = -2

b. $$\frac{q-13}{2}$$ = -8 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{q-13}{2}$$ = -8
First, undo the division by multiplying.
$$\frac{q-13}{2}$$ x 12 = -8 x 12
q-13 = -96
Then, undo the subtraction by adding.
q-13 + 13 = -96 + 13
q = -83

c. -5r + 13 = -17 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
-5r + 13 = -17
First, undo the addition by subtracting.
-5r + 13 – 13  = -17 – 13
-5r = -30
Then, undo the multiplication by dividing.
-5r ÷ -5 = -30 ÷ -5
r = 6

Question 9.
a. $$\frac{t+10}{-2}$$ = 5 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{t+10}{-2}$$ = 5
First, undo the division by multiplying.
$$\frac{t+10}{-2}$$ x (-2) = 5 x (-2)
t+10 = -10
Then, undo the addition by subtracting.
t+10 – 10 = -10 – 10
t = -20

b. $$\frac{v+8}{-2}$$ = 10 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{v+8}{-2}$$ = 10
First, undo the division by multiplying.
$$\frac{v+8}{-2}$$ x (-2) =10 x (-2)
v+8 = -20
Then, undo the addition by subtracting.
v+8-8 = -20-8
v = -28

c. -14x – 19 = 303 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
-14x – 19 = 303
First, undo the subtraction by adding.
-14x – 19  + 19 = 303 + 19
-14x = 322
Then, undo the multiplication by dividing.
-14x ÷ -14 = 322 ÷ -14
x = -23

Question 10.
a. 6z – 3 = 39 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
6z – 3 = 39
First, undo the subtraction by adding.
6z – 3 + 3 = 39 + 3
6z = 42
Then, undo the multiplication by dividing.
6z ÷ 6 = 42 ÷ 6
z = 7

b. $$\frac{45}{w}$$ – 3 = 6 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{45}{w}$$ – 3 = 6
First, undo the subtraction by adding.
$$\frac{45}{w}$$ – 3 + 3 = 6 + 3
$$\frac{45}{w}$$ = 9
Then, undo the division by multiplying.
$$\frac{45}{w}$$  x w = 9  x w
45 = 9 x w
Then, undo the multiplication by dividing.
45 ÷ 9 = 9 x w ÷ 9
w = 5

c. 9d + 4 = 31 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
9d + 4 = 31
First, undo the addition by subtracting.
9d + 4 – 4 = 31 – 4
9d = 28
Then, undo the multiplication by dividing.
9d ÷ 9 = 28 ÷ 9
d = 7

Question 11.
a. 3y + 9 = 5 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
3y + 9 = 5
First, undo the addition by subtracting.
3y + 9 – 9 = 5 – 9
3y = -4
Then, undo the multiplication by dividing.
3y ÷ 3 = -4 ÷ 3
y = -1.3

b. 12n – 2 = 4 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
12n – 2 = 4
First, undo the subtraction by adding.
12n – 2 + 2 = 4 + 2
12n = 6
Then, undo the multiplication by dividing.
12n ÷ 12 = 6 ÷ 12
n = 0.5

c. v + $$\frac{8}{9}$$ = 10 ________
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
v + $$\frac{8}{9}$$ = 10
First, undo the multiplication by dividing.
v x 9 + $$\frac{8}{9}$$ x 9 = 10 x 9
9v + 8 = 90
Then, undo the addition by subtracting.
9v + 8 – 8 = 90 – 8
9v = 82
Then, undo the multiplication by dividing.
9v ÷ 9 = 82 ÷ 9
v = $$\frac{82}{9}$$

Question 12.
a. 10 – 7y = 3 _____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
10 – 7y = 3
First, undo the addition by subtracting.
10 – 7y – 10 = 3 -10
-7y = -7
Then, undo the multiplication by dividing.
-7y ÷ -7 = -7 ÷ -7
y = 1

b. 3 – $$\frac{a}{5}$$ = 4 _____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
3 – $$\frac{a}{5}$$ = 4
First, undo the addition by subtracting.
3 – $$\frac{a}{5}$$ – 3 = 4 – 3
– $$\frac{a}{5}$$ = 1
Then, undo the division by multiplying.
– $$\frac{a}{5}$$ x 5 = 1 x 5
a = -5

c. $$\frac{m}{12}$$ = -7 ___
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{m}{12}$$ = -7
undo the division by multiplying.
$$\frac{m}{12}$$  x 12 = -7 x 12
m = -84

Question 13.
a. 5g – 2 = 10 _____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
5g – 2 = 10
First, undo the subtraction by adding.
5g – 2 + 2 = 10 + 2
5g = 12
Then, undo the multiplication by dividing.
5g ÷ 5 = 12 ÷ 5
g = 2.4

b. 28 – $$\frac{d}{70}$$ = 56 _____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
28 – $$\frac{d}{70}$$ = 56
First, undo the addition by subtracting.
28 – $$\frac{d}{70}$$ – 28  = 56 – 28
– $$\frac{d}{70}$$ = 28
Then, undo the division by multiplying.
– $$\frac{d}{70}$$ x 70 = 28 x 70
d = – 1960

c. $$\frac{r}{93}$$ = 84 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{r}{93}$$ = 84
undo the division by multiplying.
$$\frac{r}{93}$$ x 93 = 84 x 93
r = 7812

Question 14.
a. 4v + 37 = 44 _____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
4v + 37 = 44
First, undo the addition by subtracting.
4v + 37 – 37 = 44 – 37
4v = 7
Then, undo the multiplication by dividing.
4v ÷ 4 = 7 ÷ 4
v = 1.75

b. 6u – 40 = 54 ______
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
6u – 40 = 54
First, undo the subtraction by adding.
6u – 40 + 40  = 54 + 40
6u = 94
Then, undo the multiplication by dividing.
6u ÷ 6 = 94 ÷ 6
u = 15.66

c. $$\frac{6b}{14}$$ = 24 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{6b}{14}$$ = 24
First, undo the division by multiplying.
$$\frac{6b}{14}$$ x 14= 24 x 14
6b = 336
Then, undo the multiplication by dividing.
6b ÷ 6 = 336 ÷ 6
b = 56

Question 15.
a. $$\frac{a}{46}$$ = 88 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{a}{46}$$ = 88
undo the division by multiplying.
$$\frac{a}{46}$$ x 46 = 88 x 46
a = 4048

b. 83 – $$\frac{a}{27}$$ = 37 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
83 – $$\frac{a}{27}$$ = 37
First, undo the addition by subtracting.
83 – $$\frac{a}{27}$$ – 83 = 37 – 83
– $$\frac{a}{27}$$ = -46
Then, undo the division by multiplying.
– $$\frac{a}{27}$$ x 27 = -46 x 27
a = 1242

c. 5z + 80 = 45 _______
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
5z + 80 = 45
First, undo the addition by subtracting.
5z + 80 – 80 = 45 – 80
5z = -35
Then, undo the multiplication by dividing.
5z ÷ 5 = -35 ÷ 5
z = -7

Question 16.
a. 58 – $$\frac{d}{90}$$ = 93 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
58 – $$\frac{d}{90}$$ = 93
First, undo the addition by subtracting.
58 – $$\frac{d}{90}$$ – 58 = 93 -58
– $$\frac{d}{90}$$ = 35
Then, undo the division by multiplying.
– $$\frac{d}{90}$$ x 90 = 35 x 90
d = -3150

b. 30 – $$\frac{r}{95}$$ = 3 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
30 – $$\frac{r}{95}$$ = 3
First, undo the addition by subtracting.
30 – $$\frac{r}{95}$$ – 30 = 3 – 30
– $$\frac{r}{95}$$ = -27
Then, undo the division by multiplying.
– $$\frac{r}{95}$$ x 95 = -27 x 95
r = 2565

c. $$\frac{4 u}{32}$$ = 13 ____
Some problems with variables require more than one step to solve. Use the properties of equality to undo each step and find the value of the variable.
$$\frac{4 u}{32}$$ = 13
First, undo the division by multiplying.
$$\frac{4 u}{32}$$ x 32 = 13 x 32
4u = 416
Then, undo the multiplication by dividing.
4u ÷ 4 = 416 ÷ 4
u = 104

Sometimes like terms in equations have to be combined in order to solve the problem. When terms have the same variable raised to the same exponent, they can be added or subtracted. Other times, you can use the Distributive Property to combine terms.
2x + 3x = 75
5x = 75
5x ÷ 5 = 75 ÷ 5
x = 15

Using the Distributive Property to Combine Terms
2(x + 3) = 46
2x + 6 = 46
2x + 6 – 6 = 46 – 6
2x ÷ 2 = 40 ÷ 2
x = 20

Find the value of the variable in each equation by combining like terms.

Question 1.
a. 3x + 4 + 2x + 5 = 34 ____
3x + 4 + 2x + 5 = 34
5x + 9 = 34
5x + 9 – 9 = 34 – 9
5x = 25
5x  ÷ 5 = 25 ÷ 5
x = 5

b. 2(x + 1) + 4 = 12 ____
2(x + 1) + 4 = 12
2x + 2 + 4 = 12
2x + 6 = 12
2x + 6 – 6 = 12 – 6
2x = 6
2x ÷ 2 = 6 ÷ 2
x = 3

Question 2.
a. $$\frac{1}{2}$$ (x + 8) – 15 = -3 ____
$$\frac{1}{2}$$ (x + 8) – 15 = -3
x + 8 – 30 = -6
x – 22 = – 6
x -22 + 22 = -6 + 22
x = 16

b. 2x – 5 + 3x + 8 = 18 ____
2x – 5 + 3x + 8 = 18
5x + 3 = 18
5x + 3 – 3 = 18 – 3
5x = 15
5x ÷ 5 = 15 ÷ 5
x = 3

Question 3.
a. -185 = -3r – 4(-5r + 8) ___
-185 = -3r – 4(-5r + 8)
-185 = -3r + 20r – 32
-185 = 17r – 32
-185 + 32 = 17r – 32 + 32
-153 = 17r
-153 ÷ 17 = 17r ÷ 17
r = 9

b. -5t – 2(5t + 10) = 100 ___
-5t – 2(5t + 10) = 100
-5t – 10t -20 = 100
-15t – 20 = 100
-15t -20 + 20 = 100 + 20
-15t = 120
-15t ÷ 15 = 120 ÷ 15
t = -8

Question 4.
a. -4b – 4(-6b – 8) = 172 ____
-4b – 4(-6b – 8) = 172
-4b + 24b + 32 = 172
20b + 32 = 172
20b + 32 – 32 = 172 – 32
20b = 140
b = 7

b. -3p + 2(5p – 12) = -73 ____
-3p + 2(5p – 12) = -73
-3p + 10p – 24 = -73
7p -24 = -73
7p – 24 + 24 = -73 + 24
7p = -49
p = -7

Question 5.
a. -3f + 3(-3f + 5) = -81 ___
-3f + 3(-3f + 5) = -81
-3f -9f + 15 = -81
-12f + 15 = -81
-12f + 15 – 15 = -81 -15
-12f = -96
f = 8

b. -43 = -5c + 4(2c + 7) ___
-43 = -5c + 4(2c + 7)
-43 = -5c + 8c + 28
-43 = 3c + 28
-43 – 28 = 3c + 28 -28
-71 = 3c
c = 23.6666

Question 6.
a. -5s + 3(5s + 2) = 126 ____
-5s + 3(5s + 2) = 126
-5s + 15s + 6 = 126
10s + 6 – 6 = 126 – 6
10s = 120
s = 12

b. 4d + 2(4d + 7) = -106 ___
4d + 2(4d + 7) = -106
4d + 8d + 14 = -106
12d + 14 = -106
12d  + 14 – 14 = -106 – 14
12d = -120
d = -10

Question 7.
a. 103 = -2u + 3(-3u + 5) ____
103 = -2u + 3(-3u + 5)
103 = -2u – 9u + 15
103 = -11u + 15
103 – 15 = -11u
88 = -11u
u = -8

b. -2n + 2(3n + 14) = -20 ___
-2n + 2(3n + 14) = -20
-2n + 6n + 28 = -20
4n + 28 = -20
4n + 28 – 28 = -20 – 28
4n = -48
n = -12

Question 8.
a. -11 = 5y + 4(-y – 4)
-11 = 5y + 4(-y – 4)
-11 = 5y – 4y – 16
-11 = y – 16
-11 + 16 = y – 16 + 16
5 = y
y = 5

b. -5a – 2(-7a – 10) = 128 ___
-5a – 2(-7a – 10) = 128
-5a + 14a + 20 =128
9a + 20 = 128
9a + 20 – 20 = 128 – 20
9a = 108
a = 12

Question 9.
a. $$\frac{1}{2}$$(c + 5) – 10 = -4
$$\frac{1}{2}$$(c + 5) – 10 = -4
c + 5 – 20 = -8
c -15 = -8
c – 15 + 15 = -8 +15
c = 7

b. -4f + 2 (4f – 5) = -19 ____
-4f + 2 (4f – 5) = -19
-4f + 8f -10 = -19
4f – 10 = -19
4f – 10 + 10 = -19 + 10
4f = -9
f = -2.25

Question 10.
a. 2(v + 4) + 6 = 24 ___
2(v + 4) + 6 = 24
2v + 8 + 6 = 24
2v + 14 = 24
2v + 14 – 14 = 24 – 14
2v = 10
v =5

b. -9 = 6h + 3(-h – 3) ___
-9 = 6h + 3(-h – 3)
-9 = 6h -3h -9
-9 = 3h – 9
-9 + 9 = 3h – 9 + 9
0 = 3h
h = 0

Question 11.
a. -6p – 8(4p + 8) = 98 ___
-6p – 8(4p + 8) = 98
-6p – 32p -64 = 98
-38p – 64 = 98
-38p – 64 + 64 = 98 + 64
-38p = 162
p = -4.26

b. 7c + 3(3c + 5) = -103 ___
7c + 3(3c + 5) = -103
7c + 9c + 15 = -103
16c + 15 = -103
16c + 15 – 15 = -103 – 15
16c = -118
c = -7.375

Question 12.
a. -4s + 2(4s + 1) = 125 ____
-4s + 2(4s + 1) = 125
-4s + 8s + 2 = 125
4s + 2 = 125
4s + 2 – 2 = 125 – 2
4s = 123
s = 30.75

b. -3n + 3(4n + 15) = -21 ____