Students can use the Spectrum Math Grade 8 Answer Key Chapter 3 Lesson 3.3 Solving 1-Variable Equations as a quick guide to resolve any of their doubts.
Spectrum Math Grade 8 Chapter 3 Lesson 3.3 Solving 1-Variable Equations Answers Key
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal:
4 + 17 = 21 4 + 17 + 5 = 21 + 5 (26 = 26)
When the same number is subtracted from both sides of an equation, the two sides remain equal:
32 = 16 + 16 32 – 4 = 16 + 16 – 4 (28 = 28)
Use these properties to determine the value of variables:
x + 17 = 23
x + 17 – 17 = 23 – 17
x + 0 = 6 x = 6
40 – n = 19
40 – n – 40 = 19 – 40
0 – n = -29 n = 29
y – 14 = 3
y – 14 + 14 = 3 + 14
y + 0 = 17 y = 17
Find the value of the variable in each equation.
Question 1.
a. a + 12 = 25 ____
Answer: a = 13
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
a + 12 = 25
Subracting 12 on both sides
a + 12 – 12 = 25 – 12
a + 0 = 13
a = 13
b. 48 + d = 60 ____
Answer: d = 12
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
48 + d = 60
subtracting 48 on both sides
48 + d – 48 = 60 – 48
d + 0 = 12
d = 12
c. y – 19 = 18 ____
Answer: y = 37
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
y – 19 = 18
Adding 19 on both sides
y – 19 + 19 = 18 + 19
y – 0 = 37
y = 37
Question 2.
a. 31 – x = 16 ____
Answer: x = 15
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
31 – x = 16
subtracting 31 on both sides
31 – x + 31 = 16 – 31
0 – x = -15
-x = -15
x = 15
b. 11 + n = 25 ____
Answer: n = 14
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
11 + n = 25
subtractiong 11 on both sides
11 + n – 11 = 25 – 11
n = 14
c. m – 21 = 34 ____
Answer: m = 55
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
m – 21 = 34
Adding 21 on both sides
m – 21 + 21 = 34 + 21
m – 0 = 55
m = 55
Question 3.
a. 28 + b = 50 _____
Answer: b = 22
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
28 + b = 50
Subtracting 28 on both sides
28 + b – 28 = 50 – 28
b – 0 = 22
b = 22
b. p – 16 = 32 ____
Answer: p = 48
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
p – 16 = 32
Adding 16 on both sides
p – 16 + 16 = 32 + 16
p – 0 = 48
p = 48
c. t + 22 = 57 ____
Answer: t = 35
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
t + 22 = 57
Subtracting 22 on both sides
t + 22 – 22 = 57 – 22
t – 0 = 35
t = 35
Question 4.
a. 33 + c = 514 ____
Answer: c = 481
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
33 + c = 514
Subtracting 33 on both sides
33 + c – 33 = 514 – 33
c – 0 = 481
c = 481
b. e + 19 = 37 ____
Answer: e = 18
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
e + 19 = 37
Subtracting 19 on both sides
e + 19 – 19 = 37 -19
e – 0 = 18
e = 18
c. 16 + r = 0 ____
Answer: r = -16
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
16 + r = 0
Subtracting 16 on both sides
16 + r – 16 = 0 – 16
r – 0 = -16
r = -16
Question 5.
a. 52 – n = 24 ____
Answer: n = 28
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
52 – n = 24
Subtracting 52 on both sides
52 – n – 52 = 24 – 52
0 – n = -24
n = 24
b. y – 15 = 18 ____
Answer: y = 33
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
y – 15 = 18
Adding 15 on both sides
y – 15 + 15 = 18 + 15
y – 0 = 33
y = 33
c. 21 + n = 49 ____
Answer: n = 28
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
21 + n = 49
Subtracting 21 on both sides
21 + n – 21 = 49 – 21
n – 0 = 28
n = 28
Question 6.
a. m – 5 = 18 ____
Answer: m = 23
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
m – 5 = 18
Adding 5 on both sides
m – 5 + 5 = 18 + 5
m – 0 = 23
m = 23
b. 36 + s = 45 ____
Answer: s = 9
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
36 + s = 45
Subtracting 36 on both sides
36 + s – 36 = 45 – 36
s -0 = 9
s = 9
c. 21 – a = 7 ____
Answer: a = 14
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
21 – a = 7
Subtracting 7 on both sides
21 – a -7 = 7 – 7
14 – a = 0
14 = a
Question 7.
a. 17 + d = 29 ____
Answer: d = 12
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
17 + d = 29
Subtracting 17 on both sides
17 + d – 17 = 29 – 17
d – 0 = 12
d = 12
b. x – 23 = 9 ___
Answer: x = 32
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
x – 23 = 9
Adding 23 on both sides
x -23 + 23 = 9 + 23
x – 0 = 32
x = 32
c. 27 + f = 35 ____
Answer: f = 8
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
27 + f = 35
Subtracting 27 on both sides
27 + f -27 = 35 – 27
f – 0 = 8
f = 8
Question 8.
a. r – 15 = 24 ____
Answer: r = 39
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
r – 15 = 24
Adding 15 on both sides
r – 15 +15 = 24 + 15
r -0 = 39
r = 39
b. 27 – p = 3 ____
Answer: p = 24
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
27 – p = 3
Subtracting 27 on both sides
27 – p – 27 = 3 – 27
0 – p = -24
-p = -24
p = 24
c. 34 – x = 18 ____
Answer: x = 16
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
34 – x = 18
Subtracting 34 on both sides
34 – x – 34 = 18 – 34
0 – x = -16
-x = -16
x = 16
Question 9.
a. y + 12 = 20 ____
Answer: y = 8
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
y + 12 = 20
Subtracting 12 on both sides
y + 12 -12 = 20 – 12
y – 0 = 8
y = 8
b. n – 24 = 31 ___
Answer: n = 55
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
n – 24 + 24 = 31 + 24
n – 0 = 55
n = 55
c. 16 + p = 38 ___
Answer: p = 22
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables
16 + p = 38
Subtracting 16 on both sides
16 + p – 16 = 38 – 16
p – 0 = 22
p = 22
Question 10.
a. 18 + q = 25 ____
Answer: q = 7
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
18 + q = 25
Subtracting 18 on both sides
18 + q – 18 = 25 – 18
q – 0 = 7
q = 7
b. m + 17 = 32 ___
Answer: m = 15
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
m + 17 = 32
Subtracting 17 on both sides
m + 17 – 17 = 32 – 17
m + 0 = 15
m = 15
c. e + 29 = 36 ___
Answer: e = 7
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
e + 29 = 36
Subtracting 29 on both sides
e + 29 – 29 = 36 – 29
e – 0 = 7
e = 7
Question 11.
a. 39 – r = 34 ___
Answer: r = 5
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
39 – r = 34
r = 5
b. 42 + x = 56 ___
Answer: x = 14
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
42 + x = 56
Subtracting 42 on both sides
42 + x – 42 = 56 – 42
x + 0 = 14
x = 14
c. q – 21 = 35 ___
Answer: q = 56
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
q – 21 = 35
Adding 21 on both sides
q – 21 + 21 = 35 + 21
q – 0 = 56
q = 56
Question 12.
a. 18 + p = 22 ___
Answer: p = 4
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
18 + p = 22
Subtracting 18 on both sides
18 + p – 18 = 22 – 18
p – 0 = 4
p = 4
b. s – 32 = 9 ___
Answer: s = 41
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
s – 32 = 9
Adding 32 on both side
s -32 + 32 = 9 + 32
s – 0 = 41
s = 41
c. 43 + n = 49 ____
Answer: n = 6
The Addition and Subtraction Properties of Equality state that when the same number is added to both sides of on equation, the two sides remain equal. When the same number is subtracted from both sides of an equation, the two sides remain equal. Use these properties to determine the value of variables:
43 + n = 49
Subtracting 43 on both sides
43 + n – 43 = 49 – 43
n – 0 = 6
n = 6
The Multiplication and Division Properties of Equality state that when each side of the equation is multiplied by the same number, the two sides remain equal:
3 + 4 = 7 (3 + 4) × 5 = 7 × 5 (35 = 35)
When each side of the equation is divided by the same number, the two sides remain equal:
2 × 6 = 12 \(\frac{(2 \times 6)}{3}\) = \(\frac{12}{3}\) (4 = 4)
Use these properties to determine the value of variables:
n ÷ 5 = 4
n ÷ 5 × 5 = 4 × 5
n = 20
3n = 18
\(\frac{3 n}{3}\) = \(\frac{18}{3}\)
n = 6
\(\frac{60}{n}\) = 4
\(\frac{60}{n}\) = 4n or 60 = 4n
\(\frac{60}{n}\) = \(\frac{4 n}{4}\) 15 = n
Find the value of the variable in each equation.
Question 1.
a. 5b = 35 _____
Answer: b = 7
\(\frac{5 b}{5}\) = \(\frac{35}{5}\)
b = 7
b. \(\frac{60}{n}\) = 16 _______
Answer: n = 3.75
\(\frac{60 n}{60}\) = \(\frac{16}{60}\)
n = \(\frac{16}{60}\) = 3.75
c. f × 12 = 72 _____
Answer: f = 6
\(\frac{(f \times 12)}{12}\) = \(\frac{72}{12}\)
f = 6
Question 4.
a. \(\frac{n}{20}\) = 4 _____
Answer: n = \(\frac{1}{5}\)
\(\frac{n}{20}\) = \(\frac{4}{20}\)
\(\frac{n}{20}\) = \(\frac{1}{5}\)
b. a × 12 = 60 _____
Answer: a = 5
\(\frac{(a \times 12)}{12}\) = \(\frac{60}{12}\)
a = 5
c. 6p = 90 ____
Answer: p = 15
\(\frac{6 p}{6}\) = \(\frac{90}{6}\)
p = 15
Question 5.
a. x ÷ 7 = 11 ____
Answer: x = 77
x ÷ 7 = 11
x ÷ 7 x 7 = 11 x 7
x = 77
b. t ÷ 25 = 8 _______
Answer: t = 200
t ÷ 25 = 8
t ÷ 25 x 25 = 8 x 25
t = 200
c. \(\frac{x}{15}\) = 6 ____
Answer: x = 90
\(\frac{x}{15}\) = 6
\(\frac{x}{15}\) x 15 = 6 x 15
x = 90
Question 6.
a. b × 16 = 64 ____
Answer: b = 4
b x 16 ÷ 16 = 64 ÷ 16
b = 4
b. 11d = 132 ____
Answer: d = 12
\(\frac{11 d}{11}\) = \(\frac{132}{11}\)
d = 12
c. \(\frac{65}{m}\) = 5 _______
Answer: m = \(\frac{1}{13}\)
\(\frac{65 m}{65}\) = \(\frac{5}{65}\)
m = \(\frac{5}{65}\) = \(\frac{1}{13}\)
Question 7.
a. \(\frac{n}{14}\) = 3 _____
Answer: n = 42
\(\frac{n}{14}\) x 14 = 3 x 14
n = 42
b. f × 9 = 99 ____
Answer: f = 11
\(\frac{f 9 }{9}\) = \(\frac{99}{9}\)
f = 11
c. 4n = 60 ____
Answer: n =15
\(\frac{4 n }{4}\) = \(\frac{60}{4}\)
n = 15
Question 8.
a. e × 5 = 120 ___
Answer: e = 14
e x 5 ÷ 5 = 120 ÷ 5
e = 14
b. \(\frac{120}{m}\) = 10 ____
Answer: m = \(\frac{1}{12}\)
\(\frac{120 m}{120}\) = \(\frac{10}{120}\)
c. b ÷ 9 = 7 ____
Answer: b = 63
b ÷ 9 x 9 = 7 x 9
b = 63
Question 9.
a. 8t = 104 _____
Answer: t = 13
8t ÷ 8 = 104 ÷ 8
t = 13
b. \(\frac{b}{9}\) = 6 _____
Answer: b = 54
\(\frac{b}{9}\) x 9 = 6 x 9
b = 54
c. m × 18 = 54 ____
Answer: m = 3
m x 18 ÷ 18 = 54 ÷ 18
m = 3
Question 10.
a. \(\frac{a}{6}\) = 12 ____
Answer: a = 72
\(\frac{a}{6}\) x 6 = 12 x 6
a = 72
b. 7m = 84 _____
Answer: m = 12
7m ÷ 7 = 84 ÷ 7
m = 12
c. a ÷ 4 = 18 ____
Answer: a = 72
a ÷ 4 x 4 = 18 x 4
a = 72
One-variable equations can be solved by isolating the variable on one side of the equation by performing inverse operations.
Addition
t + 4 = 16
t + 4 – 4 = 16 – 4
t = 12
Subtraction
28 – r = 15
28 – r – 28 = 15 – 28
-r = -13 r = 13
Multiplication
5n = 65
5n ÷ 5 = 65 ÷ 5
n = 13
Division
72 ÷ r = 9
72 = r × r = 9 × r
72 = 9r
72 ÷ 9 = 9r ÷ 9
r = 8
Find the value of the variable in each equation.
Question 1.
a. r × 13 = 13 _____
Answer: r = 1
r x 13 ÷ 13 = 13 ÷ 13
r = 1
b. w + 18 = 22 ____
Answer: w = 4
w + 18 – 18 = 22 -18
w = 4
c. 17 × v = 153 ____
Answer: v = 9
17 x v ÷ 17 = 153 ÷ 17
v = 9
Question 2.
a. f ÷ 12 = 7 ____
Answer: f = 84
f ÷ 12 x 12 = 7 x 12
f = 84
b. y ÷ 8 = 17 ___
Answer: y = 136
y ÷ 8 x 8 = 17 x 8
y = 136
c. 24 – q = 13 ____
Answer: q = 11
24 – q – 24 = 13 -24
0 – q = – 11
q = 11
Question 3.
a. d × 7 = 35 _____
Answer: d = 5
d x 7 ÷ 7 = 35 ÷ 7
d = 7
b. t ÷ 11 = 18 ____
Answer: t = 198
t ÷ 11 x 11 = 18 x 11
t = 198
c. v + 19 = 36 _______
Answer: v = 17
v + 19 -19 = 36 -19
v – 0 = 17
v = 17
Question 4.
a. q + 8 = 16 ____
Answer: q = 8
q + 8 – 8 = 16 – 8
q – 0 = 8
q = 8
b. 66 ÷ w = 11 ____
Answer: w = 726
66 ÷ w x 66 = 11 x 66
w = 726
c. y ÷ 9 = 8 ____
Answer: y = 72
y ÷ 9 x 9 = 8 x 9
y = 72
Question 5.
a. v – 8 = 9 ____
Answer: v = 17
v – 8 + 8 = 9 + 8
v = 17
b. 17 + d = 29 ___
Answer: d = 12
17 + d – 17 = 29 – 17
d = 12
c. 4 + s = 20 ____
Answer: s = 16
4 + s – 4 = 20 -4
s = 16
Question 6.
a. 300 ÷ d = 20 ____
Answer: d = 6000
300 ÷ d x 300 = 20 x 300
d = 6000
b. 15u = 135 ____
Answer: u = 9
15u ÷ 15 = 135 ÷ 15
u = 9
c. \(\frac{x}{5}\) = 12 ____
Answer: x = 60
\(\frac{x}{5}\) x 5 = 12 x 5
x = 60
Question 7.
a. q × 3 = 27 ___
Answer: q = 9
q x 3 ÷ 3 = 27 ÷ 3
q = 9
b. 28 ÷ r = 4 ___
Answer: r = 112
28 ÷ r x 28 = 4 x 28
r = 112
c. 11x = 77 ____
Answer: x = 7
11x ÷ 11 = 77 ÷ 7
x = 11
Question 8.
a. w ÷ 4 = 13 ____
Answer: w = 42
w ÷ 4 x 4 = 13 x 4
w = 42
b. x – 16 = 20 ___
Answer: x =36
x – 16+ 16 = 20 + 16
x = 36
c. 20 – d = 13 ____
Answer: d = 7
20 – d -20 = 13 – 20
d = 7
Question 9.
a. 29 – y = 19 ___
Answer: y = 10
29 – y -29 = 19 – 29
-y = -10
y = 10
b. d × 15 = 75 ___
Answer: d =5
d x 15 ÷ 15 = 75 ÷ 15
d = 5
c. 27 ÷ t = 9 ____
Answer: t = 243
27 ÷ t x 27 = 9 x 27
t = 243
Question 10.
a. y ÷ 18 = 11 ___
Answer: y = 198
y ÷ 18 x 18 = 11 x 18
y = 198
b. w × 20 = 20 ____
Answer: w = 1
w x 20 ÷ 20 = 20 ÷ 20
w = 1
c. w + 14 = 22 ____
Answer: w = 8
w + 14 – 14 = 22 -14
w = 8
Question 11.
a. j ÷ 20 = 3 ___
Answer: j = 60
j ÷ 20 x 20 = 3 x 20
j = 60
b. 12c = 156 ____
Answer: c = 13
12c ÷ 12 = 156 ÷ 12
c = 13
c. n + 16 = 31 ____
Answer: n = 15
n + 16 – 16 = 31 – 16
n = 15
Question 12.
a. \(\frac{225}{r}\) = 15 ___
Answer: r = \(\frac{1}{15}\)
\(\frac{225 r}{225}\) = \(\frac{15}{225}\)
r = \(\frac{1}{15}\)
b. 12 – q = 8 ___
Answer: q = 4
12 – q – 12 = 8 – 12
-q = -4
q = 4
c. x – 19 = 1 ____
Answer: x = 20
x – 19 + 19 = 1 + 19
x = 20