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