Chapter 3 – Linear Equations
3.1
Answer Key
One-Step Equations and Inverse Operations
Answers 1.
𝑥 = −4
2.
𝑥 = 4.3
3.
𝑥=3
4.
𝑥=4
5.
𝑥 = 1.6
6.
𝑥 = −1
7.
𝑥 = 1 24
8.
𝑥 = 1,100
9.
𝑞=0
1
5 6
5
10. 𝑥 = 1.9001 1
11. 𝑥 = 7 1
12. 𝑥 = − 6 13. 𝑓 = 1 1
14. 𝑦 = −1 2 1
15. 𝑟 = 16 2
16. 𝑏 = 3
CK-12 Basic Algebra Concepts
1
Answer Key
Chapter 3 – Linear Equations
3.2
Applications of One-Step Equations
Answers 1.
a) 10 + 𝑛 = 25, 𝑛 = 15 10 8 𝑛 ,𝑟 𝑤
b) 𝑤 = c) 𝑟 = 2.
=
15 10 8
= 12 𝑤𝑒𝑒𝑘𝑠
a) 𝑢 = 3(8.5 + 1.25), 𝑢 = $29.25 b) 𝑐 =
𝑢 ,𝑐 12
=
$29.25 ,𝑐 12
= $2.44
c) 𝑤 = 𝑢 − $9.75, 𝑤 = $29.25 − $9.75, 𝑤 = $19.50
3.
a) (question ‘b’ from Ex C) 𝑥 =
53.5 12
𝑑𝑜𝑔𝑠
= 4.86 𝑚𝑖𝑛𝑢𝑡𝑒
b) (question ‘c’ from Ex C) 𝑥 = 53.5 − 3 = 50.5
CK-12 Basic Algebra Concepts
𝑑𝑜𝑔𝑠 𝑚𝑖𝑛𝑢𝑡𝑒
2
Chapter 3 – Linear Equations
3.3
Answer Key
Two-Step Equations and Properties of Equality
Answers 1.
Answers will vary: Like terms are terms including the power(s) of the same variable(s) Like terms: 3𝑥𝑦, 12𝑥𝑦, −7𝑥𝑦 Unlike Terms: 3𝑥𝑦, 3𝑥, 3𝑦, 3𝑥 2 𝑦
2.
Answers will vary: The coefficient is the constant (number) by which the variable(s) is/are multiplied.
3.
32𝑥
4.
13𝑥 2 + 26𝑥
5.
15𝑥𝑦 + 12𝑥
6.
8𝑎𝑏 + 9
7.
−2𝑚𝑛2 − 9𝑚𝑛 + 8
8.
−5𝑦 − 9 = 74 −5𝑦 = 74 + 9 83
𝑦 = −5 𝑦 = −16.6 9.
𝑥 = 20 ∶ 1.3(20) − 0.7(20) = 12
10. 𝑥 = 0.75 ∶ 6(. 75) − 1.3 = 3.2 3
3
3
11. 𝑥 = 2 ∶ 5 (2) − (3 (2) + 2) = 1 12. 𝑥 = −
11 4
∶ 4 ((−
8
11 ) 4
+ 3) = 1
8
2
13. 𝑞 = 1 15 ∶ 5 (1 15) − 7 = 3 1
3
1
5
2
14. 𝑥 = 3 18 ∶ 5 (3 18) + 2 = 3 1
1
3
1
5
15. 𝑠 = 1 3 ∶ 1 3 − (8 (1 3)) = 6
CK-12 Basic Algebra Concepts
3
Chapter 3 – Linear Equations
Answer Key
16. 𝑦 = −110 ∶ 0.1(−110) + 11 = 0 17. 𝑞 = 3 ∶
18. 𝑞 =
43 5
5(3)−7 12
∶
2
=3
43 5
5(( )−7) 12
2
=3
19. 𝑡 = 3 ∶ 33(3) − 99 = 0 4
4
20. 𝑝 = 6 5 ∶ 5 (6 5) − 2 = 32 21. 𝑥 = 7 ∶ 14(7) + 9(7) = 161 6 7
6 7
6 7
22. 𝑚 = ∶ 3 ( ) − 1 + 4 ( ) = 5 23. 𝑥 = 1 ∶ 8(1) + 3 = 11 24. 𝑥 = 9 ∶ 24 = 2(9) + 6 2 3
25. 𝑘 = 99 ∶ 66 = (99) 6 8
5 8
1 2
3 4
26. 𝑎 = − ∶ = (− ) + 1 27. 𝑑 = −7 ∶ 16 = −3(−7) − 5 28. $79.14 2
29. 6 3 30. $189 31. 10 ≥ 2.35 + .75𝑚 ∶ 𝑚 ≤ 10.2 𝑚𝑖𝑙𝑒𝑠 32. 300 ≥ 150 + 3𝑔 : He can invite a maximum of 50 guests
CK-12 Basic Algebra Concepts
4
Chapter 3 – Linear Equations
3.4
Answer Key
Multi-Step Equations with Like Terms
Answers 1.
𝑓=0
2.
𝑥=
3.
𝑦 = 25
4.
𝑥 + (𝑥 + 2) + (𝑥 + 4) + (𝑥 + 6) = 244 ∶ 58, 60, 62, 64
5.
2 𝑥 3
6.
3.50 = 𝑝 + 1.5𝑝 + (𝑝 + 1.4) ∶ 𝑝 = $0.60
7.
𝑙=
8.
𝑐 = 0.07𝑝 ∶ 5,389.12 = 0.07𝑝 ∶ 𝑝 = $76,987.43
15 7
+ 4 = 22 ∶ 𝑥 = 27
𝑑 5
∶𝑙=
65 5
∶ 𝑙 = 13
CK-12 Basic Algebra Concepts
5
Chapter 3 – Linear Equations
3.5
Answer Key
Distributive Property for Multi-Step Equations
Answers 1.
𝑥=9
2.
𝑤 = −22.5
3.
𝑥 = 32
4.
𝑎 = 21
5.
𝑖 = 1 15
6.
𝑣 = 48
7.
𝑝=9
8.
𝑚=1
9.
𝑛=8
1
2
2
1
4
10. 𝑣 = 5 11. 𝑏 = 13 12. 𝑣 = −
2 5
13. 𝑠 = 6 14. 𝑚 = 0 15. 𝑘 = −2
CK-12 Basic Algebra Concepts
6
Chapter 3 – Linear Equations
Answer Key
16. 𝑥 = −7 17. 𝑎 = 2 18. 𝑛 = 0 27
19. 𝑟 = 40 20. 𝑦 = 15 21. 𝑐 = 0 22. 𝑚 = 6
CK-12 Basic Algebra Concepts
7
Chapter 3 – Linear Equations
3.6
Answer Key
Equations with Variables on Both Sides
Answers 1.
𝑥=9
2.
𝑥 = −22.5
3.
𝑥 = 32
4.
𝑎=
5.
𝑡=−
6.
𝑣=
7.
𝑦=−
8.
𝑧 = − 87
9.
𝑞=−
1
7 12 34 9
141 80 82 29 32
232 15
10. 𝑏 = 1 11. 𝑥 = 0 12. 𝑣 = 3 13. 𝑘 = −2 14. 5𝑥 − 3 = 3(𝑥 + 5): 𝑥 = 9 15. 5𝑝 + 6 = 9(𝑝 − 4) − 2 ∶ 𝑝 = $11 : Sale price = $7 16. Jaime and George will have the same amount of money in apx 7 ½ weeks 17. a)The plans cost the same at 680 texts. b)At 3000 texts, the flat-rate Plan B is cheaper. 18. a)The rentals cost the same at apx 19 1/3 hrs. b)For a 24-hour time period, the $150/day plan from Modern Rental is less expensive.
CK-12 Basic Algebra Concepts
8
Chapter 3 – Linear Equations
3.7
Answer Key
Equations with Ratios and Proportions
Answers 1.
150: 3 = 50: 1
2.
150: 175 = 6: 7
3.
200: 60 = 10: 3
4.
10: 14 = 5: 7
5.
4.5: 1 = 4.5 hot dogs per minute
6.
20: 1 = 20 pounds per square inch
7.
1: 4 =1 computer per 4 students
8.
30: 1 = 30 students per teacher
9.
3: 1 = 3 meters per floor
10. 1.2: 1 = 1.2 minutes per appointment 11.
5 1
=
30 6
12. extremes: 5, 84 means: 12, 35 4
13. 𝑥 = 2 13 14. 𝑥 = 20.16 15. 𝑥 = 3
9 19
16. 𝑥 = 500 17. 𝑥 = 7,425 11
18. 𝑥 = 162 ≈ 0.068
CK-12 Basic Algebra Concepts
9
Answer Key
Chapter 3 – Linear Equations 19. 𝑥 = 0.4225 20. 𝑥 = 19.19 36
21. 𝑥 = 55 ≈ .655 22. 𝑥 = −24.75 23. 𝑛 = −23 24. $2,270 25. Mount Yukon is 5,960 meters tall 26.
2 of 15
the students have a phone less than one year old 7
13
27. The new price is 1 13 times the price of the old one. The old one is 20 as expensive as the new one. 28.
5.80 0.879
≈ 6.6, you can buy six notebooks, with some cash left over
1 2
5 8
29. 3 cups mix and 2 cups water 30. 661 nitrogen atoms 31. 35 games
CK-12 Basic Algebra Concepts
10
Chapter 3 – Linear Equations
3.8
Answer Key
Scale and Indirect Measurement Applications
Answers 1.
Similar figures are exactly the same shape, but may be difference in size or scale.
2.
Corresponding angles of similar figures are equal. Corresponding sides are proportional.
3.
Indirect measurement is the process of measuring an object by comparing it to a similar figure and evaluating the proportional sides. Indirect measurement is particularly useful when an object is difficult to measure directly. Extremely large or small objects, objects with inaccessible sides, etc. are all excellent candidates for indirect measurement.
4.
The Law of Reflection states, “The angle at which the light reflects off the mirror is the same as the angle at which it hits the mirror.” When an object is viewed in a mirror, the angles from the object to the mirror and from the mirror to the viewer are equal.
5.
60.7 inches
6.
A 1 mile : 1 mile scale map is actual size, and a cartographer would need an inordinate volume of paper to map any significant area.
7.
The tree is apx 16 ft tall
8.
Calculated at scale 1.5 in = 7ft: a) 4.5 in = 21 ft b) 2 in = 9 ft 4 in c) 2 in = 9 ft 4 in d) 1 ¼ in = 5 ft 10 in e) 1 in = 4 ft 8 in
9.
Apx 1252’ 2”
10. 155 miles 11. 4 ft 12. 16 meters 13. 1.5 14. apx 32 ft
CK-12 Basic Algebra Concepts
11
Chapter 3 – Linear Equations
3.9
Answer Key
Conversion of Decimals, Fractions, and Percent
Answers 1.
1.1%
2.
0.1%
3.
91%
4.
175%
5.
2000%
6.
16.67%
7.
20.83%
8.
85.71%
9.
157.14%
10. 13.40% 11.
10 or 91
12.
13 20
13.
4 25
14.
1 8
15.
7 8
apx
1 9
CK-12 Basic Algebra Concepts
12
Chapter 3 – Linear Equations
3.10
Answer Key
Percent Equations
Answers 1.
192
2.
$4.50
3.
0.12
4.
46
5.
40
6.
55
7.
625%
8.
3.33%
9.
5000%
10. $14.00
CK-12 Basic Algebra Concepts
13
Chapter 3 – Linear Equations
3.11
Answer Key
Percent of Change
Answers 1.
$16,125
2.
5.83%
3.
40% discount
4.
$300
5.
$10.64
6.
Store B is the better deal at 1.14 times the wholesale price. Store A charges 1.4 times the wholesale price.
7.
142 like reality shows
CK-12 Basic Algebra Concepts
14
Chapter 3 – Linear Equations
3.12
Answer Key
Formulas for Problem Solving
Answers 1
13
1.
(5)(8) (ℎ − ) = 30 ∶ ℎ = 𝑓𝑡 𝑜𝑟 1′1" 3 12
2.
36 = .9(. 8𝑥) ∶ 𝑥 = $50.00
3.
0.06 sec delay
4.
$650 on vacation
5.
13.11 𝑐𝑚2
6.
1.91𝐺𝐵 𝑐𝑚2 3 2 4
𝜋( ) (5)
7.
𝑉=
8.
39,370 in
9.
2,172.61 m
3
15
= 16 𝜋 𝑖𝑛3 ≈ 2.95 𝑖𝑛3
10. 12 cm
CK-12 Basic Algebra Concepts
15
