Prealgebra and Introductory Algebra

Richard N. Aufmann, Joanne S. Lockwood

Chapter 5

Solving Equations - all with Video Answers

Educators

Section 1

Introduction to Equations

Problem 1

Label each of the following as either an expression or an equation.
a. $3 x+7=9$
b. $3 x+7$
c. $4-6(y+5)$
d. $a+b=8$
e. $a+b-8$

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Problem 2

What is the solution of the equation $x=8 ?$ Use your answer to explain why the goal in solving equations is to get the variable alone on one side of the equation.

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Problem 3

Which of the following are equations of the form $x+a=b ?$ If an equation is of the form $x+a=b,$ what would you do to solve the equation?
(i) $d+7.8=-9.2$
(ii) $0.3=t+1.4$
(iii) $-9=3 y$
(iv) $-8+c=-5.6$

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Problem 4

Which of the following are equations of the form $a x=b ?$ If an equation is of the form $a x=b,$ what would you do to solve the equation?
(i) $3 y=-12$
(ii) $2.4=0.6 d$
(iii) $-5=z-10$
(iv) $-8 c=-56$

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Problem 5

Keith and Jennifer started at the same time and rode toward each other on a straight road. When they met, Keith had traveled 15 mi and Jennifer had traveled 10 mi. Who had the greater average speed?

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Problem 6

Suppose you have a powerboat with the throttle set to move the boat at 8 mph in calm water. The rate of the current of a river is $4 \mathrm{mph}$.
a. What is the speed of the boat when traveling on this river with the current?
b. What is the speed of the boat when traveling on this river against the current?

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Problem 7

Is 4 a solution of $2 x=8 ?$

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Problem 8

Is 3 a solution of $y+4=7 ?$

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Problem 9

Is $-1$ a solution of $2 b-1=3 ?$

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Problem 10

Is $-2$ a solution of $3 a-4=10 ?$

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Problem 11

Is 1 a solution of $4-2 m=3 ?$

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Problem 12

Is 2 a solution of $7-3 n=2 ?$

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Problem 13

Is 5 a solution of $2 x+5=3 x ?$

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Problem 14

Is 4 a solution of $3 y-4=2 y ?$

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Problem 15

Is $-2$ a solution of $3 a+2=2-a ?$

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Problem 16

Is 3 a solution of $z^{2}+1=4+3 z ?$

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Problem 17

Is 2 a solution of $2 x^{2}-1=4 x-1 ?$

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Problem 18

Is $-1$ a solution of $y^{2}-1=4 y+3 ?$

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Problem 19

Is $\frac{1}{2}$ a solution of $4 y+1=3 ?$

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Problem 20

Is $\frac{2}{5}$ a solution of $5 m+1=10 m-3 ?$

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Problem 21

Is $\frac{3}{4}$ a solution of $8 x-1=12 x+3 ?$

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Problem 22

If $A$ is a fixed number such that $A<0,$ is a solution of the equation $5 x=A$ positive or negative?

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Problem 23

Without solving the equation $x-\frac{11}{16}=\frac{19}{24}$, determine whether $x$ is less than or greater than
$\frac{19}{24} .$ Explain your answer.

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Problem 24

Without solving the equation $x+\frac{13}{15}=-\frac{21}{43}$, determine whether $x$ is less than or greater than $-\frac{21}{43} .$ Explain your answer.

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Problem 25

Solve and check.
$$x+5=7$$

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Problem 26

Solve and check.
$$y+3=9$$

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Problem 27

Solve and check.
$$b-4=11$$

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Problem 28

Solve and check.
$$z-6=10$$

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Problem 29

Solve and check.
$$2+a=8$$

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Problem 30

Solve and check.
$$5+x=12$$

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Problem 31

Solve and check.
$$n-5=-2$$

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Problem 32

Solve and check.
$$x-6=-5$$

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Problem 33

Solve and check.
$$b+7=7$$

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Problem 34

Solve and check.
$$y-5=-5$$

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Problem 35

Solve and check.
$$z+9=2$$

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Problem 36

Solve and check.
$$n+11=1$$

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Problem 37

Solve and check.
$$10+m=3$$

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Problem 38

Solve and check.
$$8+x=5$$

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Problem 39

Solve and check.
$$9+x=-3$$

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Problem 40

Solve and check.
$$10+y=-4$$

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Problem 41

Solve and check.
$$2=x+7$$

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Problem 42

Solve and check.
$$-8=n+1$$

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Problem 43

Solve and check.
$$4=m-11$$

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Problem 44

Solve and check.
$$-6=y-5$$

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Problem 45

Solve and check.
$$12=3+w$$

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Problem 46

Solve and check.
$$-9=5+x$$

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Problem 47

Solve and check.
$$4=-10+b$$

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Problem 48

Solve and check.
$$-7=-2+x$$

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Problem 49

Solve and check.
$$m+\frac{2}{3}=-\frac{1}{3}$$

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Problem 50

Solve and check.
$$c+\frac{3}{4}=-\frac{1}{4}$$

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Problem 51

Solve and check.
$$x-\frac{1}{2}=\frac{1}{2}$$

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Problem 52

Solve and check.
$$x-\frac{2}{5}=\frac{3}{5}$$

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Problem 53

Solve and check.
$$\frac{5}{8}+y=\frac{1}{8}$$

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Problem 54

Solve and check.
$$\frac{4}{9}+a=-\frac{2}{9}$$

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Problem 55

Solve and check.
$$-\frac{5}{6}=x-\frac{1}{4}$$

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Problem 56

Solve and check.
$$-\frac{1}{4}=c-\frac{2}{3}$$

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Problem 57

Solve and check.
$$d+1.3619=2.0148$$

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Problem 58

Solve and check.
$$w+2.932=4.801$$

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Problem 59

Solve and check.
$$6.149=-3.108+z$$

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Problem 60

Solve and check.
$$5.237=-2.014+x$$

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Problem 61

Solve and check.
$$5 x=-15$$

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Problem 62

Solve and check.
$$4 y=-28$$

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Problem 63

Solve and check.
$$3 b=0$$

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Problem 64

Solve and check.
$$2 a=0$$

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Problem 65

Solve and check.
$$-3 x=6$$

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Problem 66

Solve and check.
$$-5 m=20$$

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Problem 67

Solve and check.
$$-\frac{1}{6} n=-30$$

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Problem 68

Solve and check.
$$20=\frac{1}{4} c$$

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Problem 69

Solve and check.
$$0=-5 x$$

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Problem 70

Solve and check.
$$0=-8 a$$

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Problem 71

Solve and check.
$$\frac{x}{3}=2$$

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Problem 72

Solve and check.
$$\frac{x}{4}=3$$

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Problem 73

Solve and check.
$$-\frac{y}{2}=5$$

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Problem 74

Solve and check.
$$-\frac{b}{3}=6$$

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Problem 75

Solve and check.
$$\frac{3}{4} y=9$$

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Problem 76

Solve and check.
$$\frac{2}{5} x=6$$

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Problem 77

Solve and check.
$$-\frac{2}{3} d=8$$

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Problem 78

Solve and check.
$$-\frac{3}{5} m=12$$

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Problem 79

Solve and check.
$$\frac{2 n}{3}=0$$

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Problem 80

Solve and check.
$$\frac{5 x}{6}=0$$

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Problem 81

Solve and check.
$$\frac{-3 z}{8}=9$$

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Problem 82

Solve and check.
$$\frac{3 x}{4}=2$$

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Problem 83

Solve and check.
$$\frac{2}{9}=\frac{2}{3} y$$

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Problem 84

Solve and check.
$$-\frac{6}{7}=-\frac{3}{4} b$$

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Problem 85

Solve and check.
$$\frac{x}{1.46}=3.25$$

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Problem 86

Solve and check.
$$\frac{z}{2.95}=-7.88$$

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Problem 87

Solve and check.
$$3.47 a=7.1482$$

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Problem 88

Solve and check.
$$2.31 m=2.4255$$

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Problem 89

Solve and check.
$$2 m+5 m=49$$

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Problem 90

Solve and check.
$$5 x+2 x=14$$

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Problem 91

Solve and check.
$$3 n+2 n=20$$

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Problem 92

Solve and check.
$$7 d-4 d=9$$

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Problem 93

Solve and check.
$$10 y-3 y=21$$

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Problem 94

Solve and check.
$$2 x-5 x=9$$

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Problem 95

Suppose $y$ is a positive integer. Determine whether $x$ is positive or negative.
$$15 x=y$$

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Problem 96

Suppose $y$ is a positive integer. Determine whether $x$ is positive or negative.
$$-6 x=y$$

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Problem 97

Suppose $y$ is a positive integer. Determine whether $x$ is positive or negative.
$$-\frac{1}{4} x=y$$

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Problem 98

Suppose $y$ is a positive integer. Determine whether $x$ is positive or negative.
$$\frac{2}{9} x=-y$$

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Problem 99

Morgan and Emma ride their bikes from Morgan's house to the store. Morgan begins biking 5 min before Emma begins. Emma bikes faster than Morgan and catches up with her just as they reach the store.
a. Is the distance biked by Emma less than, equal to, or greater than the distance biked by Morgan?
b. Is the time spent biking by Emma less than, equal to, or greater than the time spent biking by Morgan?

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Problem 100

See the news clipping at the right. Find the time it will take the high-speed train to travel between the two cities. Round to the nearest tenth of an hour.

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Problem 101

It takes a hospital dietician 40 min to drive from home to the hospital, a distance of $20 \mathrm{mi} .$ What is the dietician's average rate of speed?

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Problem 102

As part of a training program for the Boston Marathon, a runner wants to build endurance by running at a rate of 9 mph for 20 min. How far will the runner travel in that time period?

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Problem 103

Marcella leaves home at 9: 00 A.M. and drives to school, arriving at 9: 45 A.M. If the distance between home and school is $27 \mathrm{mi}$, what is Marcella's average rate of speed?

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Problem 104

The Ride for Health Bicycle Club has chosen a 36 -mile course for this Saturday's ride. If the riders plan on averaging 12 mph while they are riding, and they have a 1-hour lunch break planned, how long will it take them to complete the trip?

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Problem 105

Palmer's average running speed is $3 \mathrm{km} / \mathrm{h}$ faster than his walking speed. If Palmer can run around a 30 -kilometer course in $2 \mathrm{h}$, how many hours would it take for Palmer to walk the same course?

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Problem 106

A shopping mall has a moving sidewalk that takes shoppers from the shopping area to the parking garage, a distance of 250 ft. If your normal walking rate is $5 \mathrm{ft} / \mathrm{s}$ and the moving sidewalk is traveling at $3 \mathrm{ft} / \mathrm{s}$, how many seconds would it take for you to walk from one end of the moving sidewalk to the other end?

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Problem 107

Two joggers start at the same time from opposite ends of an 8 -mile jogging trail and begin running toward each other. One jogger is running at a rate of $5 \mathrm{mph}$, and the other jogger is running at a rate of 7 mph. How long, in minutes, after they start will the two joggers meet?

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Problem 108

See the news clipping at the right. Two sQubas are on opposite sides of a lake $1.6 \mathrm{mi}$ wide. They start toward each other at the same time, one traveling on the surface of the water and the other traveling underwater. In how many minutes will the sQuba traveling on the surface of the water be directly above the sQuba traveling underwater? Assume they are traveling at top speed.

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Problem 109

Two cyclists start from the same point at the same time and move in opposite directions. One cyclist is traveling at $8 \mathrm{mph},$ and the other cyclist is traveling at $9 \mathrm{mph}$. After 30 min, how far apart are the two cyclists?

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Problem 110

Petra and Celine can paddle their canoe at a rate of $10 \mathrm{mph}$ in calm water. How long will it take them to travel 4 mi against the 2-mile-per-hour current of the river?

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Problem 111

At 8: 00 A.M., a train leaves a station and travels at a rate of 45 mph. At 9: 00 A.M., a second train leaves the same station on the same track and travels in the direction of the first train at a speed of 60 mph. At 10: 00 A.M., how far apart are the two trains?

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Problem 112

Solve.
$$\frac{3 y-8 y}{7}=15$$

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Problem 113

Solve.
$$\frac{2 m+m}{5}=-9$$

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Problem 114

Solve.
$$\frac{1}{\frac{1}{x}}+8=-19$$

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Problem 115

Solve.
$$\frac{1}{\frac{1}{x}}=5$$

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Problem 116

Solve.
$$
\frac{5}{\frac{7}{a}}-\frac{3}{\frac{7}{a}}=6
$$

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Problem 117

Solve.
$$
\frac{4}{\frac{3}{b}}=8
$$

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Problem 118

Make up an equation of the form $a x=b$ that has $-2$ as a solution.

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Problem 119

Make up an equation of the form $x+a=b$ that has 2 as a solution.

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Problem 120

Use the numbers $5,10$, and 15 to fill in the boxes in the equation $x+\square=\square-\square$
a. What is the largest solution possible?
b. What is the smallest solution possible?

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Problem 121

Two numbers form a "two-pair" if the sum of their reciprocals equals $2 .$ For example, $\frac{8}{15}$ and 8 are a two-pair because $\frac{15}{8}+\frac{1}{8}=2 .$ If two numbers $a$ and $b$ form a two-pair, and $a=\frac{7}{3},$ what is the value of $b ?$

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