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 ____
Answer: n = 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.
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 ___
Answer: a = 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.
\(\frac{a}{3}\) – 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 ___
Answer: b = 36
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 ____
Answer: p = 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.
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 ____
Answer: b = 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.
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 ____
Answer: r = 120
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 ____
Answer: m = 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.
\(\frac{m}{16}\) + 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 ___
Answer: n = 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.
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 ____
Answer: s = 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.
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 ____
Answer: a = 81
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 ____
Answer: d = 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.
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 ___
Answer: m = 144
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 ___
Answer: a = 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.
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
Answer: e = 120
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 ____
Answer: i = 272
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 ____
Answer: p = 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.
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 ___
Answer: n = 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.
\(\frac{n}{3}\) + 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 _____
Answer: b = 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.
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 ____
Answer: s = 60
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 ____
Answer:  x = 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.
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 ____
Answer: a = 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.
\(\frac{a}{3}\) – 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 ____
Answer: r = 18
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 ___
Answer: x = 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.
\(\frac{x}{7}\) + 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 ___
Answer:
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 ____
Answer: n = 70
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 _____
Answer: a = 96
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 _____
Answer: p = 12
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 ____
Answer: n = 70
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 ____
Answer: a = 96
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 _______
Answer: p = 12
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 ____
Answer: a = 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.
\(\frac{a}{10}\) + 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 ____
Answer: c = -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.
\(\frac{c}{2}\) + 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 ____
Answer: e = -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.
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 ___
Answer: g = 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.
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 ___
Answer: h = -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.
\(\frac{h-10}{2}\) = -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 ___
Answer: j = 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.
\(\frac{j-5}{2}\) = 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 _______
Answer: f = 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.
-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 _______
Answer: n = -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.
\(\frac{9+n}{3}\) = 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 _______
Answer: p = -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.
\(\frac{-5+p}{22}\) = -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 ____
Answer: q = 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.
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 ____
Answer: s = -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 ____
Answer: u = -21
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 ____
Answer: w = 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.
\(\frac{-4+w}{2}\) = 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 ____
Answer: y = 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.
-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 ____
Answer:  b = -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.
\(\frac{b}{4}\) + 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 ____
Answer: d = 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.
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 ____
Answer: f = 36
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 ____
Answer: h = -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.
\(\frac{h+11}{3}\) = -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 ____
Answer: j = -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.
\(\frac{j-10}{3}\) = -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 ____
Answer:
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 ____
Answer: m = 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.
\(\frac{m}{16}\) – 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 ____
Answer: o = -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.
-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 ____
Answer: q = -83
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 ____
Answer:  r = 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.
-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 ____
Answer: t = -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.
\(\frac{t+10}{-2}\) = 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 ____
Answer: v = -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.
\(\frac{v+8}{-2}\) = 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 ____
Answer: x = -23
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 ____
Answer: z = 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.
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 ____
Answer: w = 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.
\(\frac{45}{w}\) – 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 ____
Answer: d = 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.
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 ___
Answer: y = -1.3
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.
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 ___
Answer: n = 0.5
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.
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 ________
Answer:
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 _____
Answer: y = 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.
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 _____
Answer: a = -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.
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 ___
Answer: m = -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.
\(\frac{m}{12}\) = -7
undo the division by multiplying.
\(\frac{m}{12}\)  x 12 = -7 x 12
m = -84

Question 13.
a. 5g – 2 = 10 _____
Answer: g = 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.
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 _____
Answer: d = – 1960
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 ____
Answer: r = 7812
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 _____
Answer: v = 1.75
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 ______
Answer:  u = 15.66
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 ____
Answer: b = 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.
\(\frac{6b}{14}\) = 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 ____
Answer: a = 4048
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 ____
Answer: a = 1242
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 _______
Answer: z = -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.
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 ____
Answer: d = -3150
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 ____
Answer: r = 2565
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 ____
Answer: u = 104
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.
Adding or Subtracting Like 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 ____
Answer:  x = 5
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 ____
Answer: x = 3
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 ____
Answer: x =16
\(\frac{1}{2}\) (x + 8) – 15 = -3
x + 8 – 30 = -6
x – 22 = – 6
x -22 + 22 = -6 + 22
x = 16

b. 2x – 5 + 3x + 8 = 18 ____
Answer: x = 3
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) ___
Answer: r = 9
-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 ___
Answer: t = -8
-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 ____
Answer:  b = 7
-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 ____
Answer: p = -7
-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 ___
Answer: f = 8
-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) ___
Answer: c = 23.6666
-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 ____
Answer: s = 12
-5s + 3(5s + 2) = 126
-5s + 15s + 6 = 126
10s + 6 – 6 = 126 – 6
10s = 120
s = 12

b. 4d + 2(4d + 7) = -106 ___
Answer: d = -10
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) ____
Answer: u = -8
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 ___
Answer: n = -12
-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)
Answer: y = 5
-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 ___
Answer: a = 12
-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
Answer: c =7
\(\frac{1}{2}\)(c + 5) – 10 = -4
c + 5 – 20 = -8
c -15 = -8
c – 15 + 15 = -8 +15
c = 7

b. -4f + 2 (4f – 5) = -19 ____
Answer: f = -2.25
-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 ___
Answer: v =5
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) ___
Answer:
-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 ___
Answer: p = -4.26
-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 ___
Answer: c = -7.375
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 ____
Answer: s = 30.75
-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 ____
Answer: n = -7.33
-3n + 3(4n + 15) = -21
-3n + 12n + 45 = -21
9n + 45 = -21
9n + 45 – 45 = -21 – 45
9n = -66
n = -7.33

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