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2: Equations and Problem Solving

# 2: Equations and Problem Solving

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98

Chapter 3 • Equations, Inequalities, and Problem Solving

✔ Check

3x ϩ 1 ϭ 7

3122 ϩ 1 ՘ 7

6ϩ1՘7

7ϭ7

Now we know that the solution set is 526.

Classroom Example

Solve 7m Ϫ 3 ϭ 11.

EXAMPLE 2

Solve 5x Ϫ 6 ϭ 14.

Solution

5x Ϫ 6 ϭ 14

5x Ϫ 6 ϩ 6 ϭ 14 ϩ 6

5x ϭ 20

5x 20

ϭ

5

5

Divide both sides by 5

xϭ4

✔ Check

5x Ϫ 6 ϭ 14

5(4) Ϫ 6 ՘ 14

20 Ϫ 6 ՘ 14

14 ϭ 14

The solution set is {4}.

Classroom Example

Solve 5Ϫ2c ϭ 13.

EXAMPLE 3

Solve 4 Ϫ 3a ϭ 22.

Solution

4 Ϫ 3a ϭ 22

4 Ϫ 3a Ϫ 4 ϭ 22 Ϫ 4

Ϫ3a ϭ 18

18

Ϫ3a

ϭ

Ϫ3

Ϫ3

a ϭ Ϫ6

Subtract 4 from both sides

Divide both sides by Ϫ3

✔ Check

4 Ϫ 3a ϭ 22

4 Ϫ 31Ϫ62 ՘ 22

4 ϩ 18 ՘ 22

22 ϭ 22

The solution set is {Ϫ6}.

Notice that in Examples 1, 2, and 3, we used the addition-subtraction property first, and

then we used the multiplication-division property. In general, this sequence of steps provides

the easiest method for solving such equations. Perhaps you should convince yourself of that

fact by doing Example 1 again, but this time use the multiplication-division property first and

3.2 • Equations and Problem Solving

Classroom Example

Solve 21 ϭ 4x ϩ 8.

99

Solve 19 ϭ 2n ϩ 4.

EXAMPLE 4

Solution

19 ϭ 2n ϩ 4

19 Ϫ 4 ϭ 2n ϩ 4 Ϫ 4

15 ϭ 2n

15

2n

ϭ

2

2

Subtract 4 from both sides

Divide both sides by 2

15

ϭn

2

✔ Check

19 ϭ 2n ϩ 4

19 ՘ 2 a

15

bϩ4

2

19 ՘ 15 ϩ 4

19 ϭ 19

The solution set is e

15

f.

2

Word Problems

In the last section of Chapter 2, we translated English phrases into algebraic expressions. We

are now ready to expand that concept and translate English sentences into algebraic equations. Such translations allow us to use the concepts of algebra to solve word problems. Let’s

consider some examples.

Classroom Example

A certain number added to 14 yields

a sum of 32. What is the number?

EXAMPLE 5

A certain number added to 17 yields a sum of 29. What is the number?

Solution

Let n represent the number to be found. The sentence “A certain number added to 17 yields

a sum of 29” translates into the algebraic equation 17 ϩ n ϭ 29. We can solve this equation.

17 ϩ n ϭ 29

17 ϩ n Ϫ 17 ϭ 29 Ϫ 17

n ϭ 12

The solution is 12, which is the number asked for in the problem.

We often refer to the statement “let n represent the number to be found” as declaring the

variable. We need to choose a letter to use as a variable and indicate what it represents for a

specific problem. This may seem like an unnecessary exercise, but as the problems become

more complex, the process of declaring the variable becomes more important. We could solve

a problem like Example 5 without setting up an algebraic equation; however, as problems

increase in difficulty, the translation from English into an algebraic equation becomes a key

issue. Therefore, even with these relatively simple problems we need to concentrate on the

translation process.

100

Chapter 3 • Equations, Inequalities, and Problem Solving

Classroom Example

Five years ago Erik was 9 years old.

How old is he now?

EXAMPLE 6

Six years ago Bill was 13 years old. How old is he now?

Solution

Let y represent Bill’s age now; therefore, y Ϫ 6 represents his age 6 years ago. Thus

y Ϫ 6 ϭ 13

y Ϫ 6 ϩ 6 ϭ 13 ϩ 6

y ϭ 19

Bill is presently 19 years old.

Classroom Example

Dawn worked 12 hours on Friday

and earned \$111. How much did she

earn per hour?

EXAMPLE 7

Betty worked 8 hours Saturday and earned \$60. How much did she earn per hour?

Solution A

Let x represent the amount Betty earned per hour. The number of hours worked times the

wage per hour yields the total earnings. Thus

8x ϭ 60

8x

60

ϭ

8

8

x ϭ 7.50

Betty earned \$7.50 per hour.

Solution B

Let y represent the amount Betty earned per hour. The wage per hour equals the total wage

divided by the number of hours. Thus

60

8

y ϭ 7.50

Betty earned \$7.50 per hour.

Sometimes we can use more than one equation to solve a problem. In Solution A, we set

up the equation in terms of multiplication; whereas in Solution B, we were thinking in terms

of division.

Classroom Example

Ashley paid \$234 for a DVD player

and 4 DVD movies. The DVD player

cost eight times as much as one DVD

movie. Find the cost of the DVD

player and the cost of one DVD

movie.

EXAMPLE 8

Kendall paid \$244 for a CD player and six CDs. The CD player cost ten times as much as

one CD. Find the cost of the CD player and the cost of one CD.

Solution

Let d represent the cost of one CD. Then the cost of the CD player is represented by 10d, and

the cost of six CDs is represented by 6d. The total cost is \$244, so we can proceed as follows:

Cost of CD player ϩ Cost of six CDs ϭ \$244

10d

ϩ

6d

ϭ 244

3.2 • Equations and Problem Solving

101

Solving this equation, we have

16d ϭ 244

d ϭ 15.25

The cost of one CD is \$15.25, and the cost of the CD player is 10(15.25) or \$152.50.

Classroom Example

The cost of a ten-day trail ride

cost included \$950 for the adventure

and an amount for 3 nights of lodging at the camp prior to the start of

the adventure. Find the cost per night

of the lodging.

EXAMPLE 9

The cost of a five-day vacation cruise package was \$534. This cost included \$339 for the

cruise and an amount for 2 nights of lodging on shore. Find the cost per night of the lodging.

Solution

Let n represent the cost for one night of lodging; then 2n represents the total cost of lodging.

Thus the cost for the cruise and lodging is the total cost of \$534. We can proceed as follows

Cost of cruise ϩ Cost of lodging ϭ \$534

ϩ

339

2n

ϭ

534

We can solve this equation:

339 ϩ 2n ϭ 534

2n ϭ 195

2n 195

ϭ

2

2

n ϭ 97.50

The cost of lodging per night is \$97.50.

Concept Quiz 3.2

For Problems 1–5, answer true or false.

1. Only one property of equality is necessary to solve any equation.

2. Substituting the solution into the original equation to obtain a true numerical statement

can be used to check potential solutions.

3. The statement “let x represent the number” is referred to as checking the variable.

4. Sometimes there can be two approaches to solving a word problem.

1

5. To solve the equation, x Ϫ 2 ϭ 7, you could begin by either adding 2 to both

3

sides of the equation or by multiplying both sides of the equation by 3.

For Problems 6–10, match the English sentence with its algebraic equation.

6.

7.

8.

9.

Three added to a number is 24.

The product of 3 and a number is 24.

Three less than a number is 24.

The quotient of a number and three is 24.

10. A number subtracted from 3 is 24.

3x ϭ 24

3 Ϫ x ϭ 24

x ϩ 3 ϭ 24

x Ϫ 3 ϭ 24

x

E. ϭ 24

3

A.

B.

C.

D.

Problem Set 3.2

For Problems 1– 40, solve each equation. (Objective 1)

5. 3x Ϫ 1 ϭ 23

6. 2x Ϫ 5 ϭ 21

1. 2x ϩ 5 ϭ 13

2. 3x ϩ 4 ϭ 19

7. 4n Ϫ 3 ϭ 41

8. 5n Ϫ 6 ϭ 19

3. 5x ϩ 2 ϭ 32

4. 7x ϩ 3 ϭ 24

9. 6y Ϫ 1 ϭ 16

10. 4y Ϫ 3 ϭ 14

102

Chapter 3 • Equations, Inequalities, and Problem Solving

11. 2x ϩ 3 ϭ 22

12. 3x ϩ 1 ϭ 21

13. 10 ϭ 3t Ϫ 8

14. 17 ϭ 2t ϩ 5

15. 5x ϩ 14 ϭ 9

16. 4x ϩ 17 ϭ 9

17. 18 Ϫ n ϭ 23

18. 17 Ϫ n ϭ 29

19. Ϫ3x ϩ 2 ϭ 20

20. Ϫ6x ϩ 1 ϭ 43

21. 7 ϩ 4x ϭ 29

22. 9 ϩ 6x ϭ 23

23. 16 ϭ Ϫ2 Ϫ 9a

24. 18 ϭ Ϫ10 Ϫ 7a

25. Ϫ7x ϩ 3 ϭ Ϫ7

26. Ϫ9x ϩ 5 ϭ Ϫ18

27. 17 Ϫ 2x ϭ Ϫ19

28. 18 Ϫ 3x ϭ Ϫ24

29. Ϫ16 Ϫ 4x ϭ 9

30. Ϫ14 Ϫ 6x ϭ 7

31. Ϫ12t ϩ 4 ϭ 88

32. Ϫ16t ϩ 3 ϭ 67

33. 14y ϩ 15 ϭ Ϫ33

34. 12y ϩ 13 ϭ Ϫ15

35. 32 Ϫ 16n ϭ Ϫ8

36. Ϫ41 ϭ 12n Ϫ 19

37. 17x Ϫ 41 ϭ Ϫ37

38. 19y Ϫ 53 ϭ Ϫ47

39. 29 ϭ Ϫ7 Ϫ 15x

40. 49 ϭ Ϫ5 Ϫ 14x

For each of the following problems, (a) choose a variable

and indicate what it represents in the problem, (b) set up an

equation that represents the situation described, and

(c) solve the equation. (Objective 2)

41. Twelve added to a certain number is 21. What is the

number?

42. A certain number added to 14 is 25. Find the number.

43. Nine subtracted from a certain number is 13. Find the

number.

44. A certain number subtracted from 32 is 15. What is the

number?

45. Suppose that two items cost \$43. If one of the items

costs \$25, what is the cost of the other item?

46. Eight years ago Rosa was 22 years old. Find Rosa’s

present age.

47. Six years from now, Nora will be 41 years old. What is

her present age?

48. Chris bought eight pizzas for a total of \$50. What was

the price per pizza?

49. Chad worked 6 hours Saturday for a total of \$39. How

much per hour did he earn?

50. Jill worked 8 hours Saturday at \$5.50 per hour. How

much did she earn?

51. If 6 is added to three times a certain number, the result

is 24. Find the number.

52. If 2 is subtracted from five times a certain number, the

result is 38. Find the number.

53. Nineteen is 4 larger than three times a certain number.

Find the number.

54. If nine times a certain number is subtracted from 7, the

result is 52. Find the number.

55. Dress socks cost \$2.50 a pair more than athletic socks.

Randall purchased one pair of dress socks and six pairs

of athletic socks for \$21.75. Find the price of a pair of

dress socks.

56. Together, a calculator and a mathematics textbook cost

\$85 in the college bookstore. The textbook price is \$45

more than the price of the calculator. Find the price of

the textbook.

57. The rainfall in June was 11.2 inches. This was 1 inch

less than twice the rainfall in July. Find the amount of

rainfall in inches for July.

58. Lunch at Joe’s Hamburger Stand costs \$1.75 less

than lunch at Jodi’s Taco Palace. A student spent his

weekly lunch money, \$24.50, eating four times at Jodi’s

and one time at Joe’s. Find the cost of lunch at Jodi’s

Taco Palace.

59. If eight times a certain number is subtracted from 27,

the result is 3. Find the number.

60. Twenty is 22 less than six times a certain number. Find

the number.

61. A jeweler has priced a diamond ring at \$550. This price

represents \$50 less than twice the cost of the ring to the

jeweler. Find the cost of the ring to the jeweler.

62. Todd is following a 1750-calorie-per-day diet plan. This

plan permits 650 calories less than twice the number of

calories permitted by Lerae’s diet plan. How many calories are permitted by Lerae’s plan?

63. The length of a rectangular floor is 18 meters. This

length is 2 meters less than five times the width of the

floor. Find the width of the floor.

64. An executive is earning \$145,000 per year. This is

\$15,000 less than twice her salary 4 years ago. Find her

salary 4 years ago.

65. In the year 2000 it was estimated that there were

874 million speakers of the Chinese Mandarin language. This was 149 million less than three times the

speakers of the English language. By this estimate how

many million speakers of the English language were

there in the year 2000?

66. A bill from the limousine company was \$510. This

included \$150 for the service and \$80 for each hour of

use. Find the number of hours that the limousine was

used.

3.3 • More on Solving Equations and Problem Solving

67. Robin paid a \$454 bill for a car DVD system. This

included \$379 for the DVD player and \$60 an hour for

installation. Find the number of hours it took to install

the DVD system.

68. Tracy received a bill with cell phone use charges of

\$136.74. Included in the \$136.74 was a charge of

103

\$39.99 for the monthly plan and a charge for 215 extra

minutes. How much is Tracy being charged for each

extra minute?

Additional word problems can be found in Appendix B.

All of the problems in the Appendix marked as (3.2) are

appropriate for this section.

Thoughts Into Words

69. Give a step-by-step description of how you would solve

the equation 17 ϭ Ϫ3x ϩ 2.

70. What does the phrase “declare a variable” mean when

solving a word problem?

71. Suppose that you are helping a friend with his homework, and he solves the equation 19 ϭ 14 Ϫ x like this:

19 ϩ x ϭ 14

19 ϩ x Ϫ 19 ϭ 14 Ϫ 19

x ϭ Ϫ5

The solution set is {Ϫ5} .

Does he have the correct solution set? What would you

say to him about his method of solving the equation?

19 ϭ 14 Ϫ x

19 ϩ x ϭ 14 Ϫ x ϩ x

1. False

2. True

3. False

4. True

3.3

5. True

6. C

7. A

8. D

9. E

10. B

More on Solving Equations and Problem Solving

OBJECTIVES

1

Solve ﬁrst-degree equations by simplifying both sides and then applying

properties of equality

2

Solve ﬁrst-degree equations that are contradictions

3

Solve ﬁrst-degree equations that are identities

4

Solve word problems that represent several quantities in terms of the

same variable

5

Solve word problems involving geometric relationships

As equations become more complex, we need additional tools to solve them. So we need to

organize our work carefully to minimize the chances for error. We will begin this section with

some suggestions for solving equations, and then we will illustrate a solution format that is

effective.

We can summarize the process of solving first-degree equations of one variable with the

following three steps.

Step 1

Simplify both sides of the equation as much as possible.

104

Chapter 3 • Equations, Inequalities, and Problem Solving

Use the addition or subtraction property of equality to isolate a term that contains

the variable on one side and a constant on the other side of the equal sign.

Step 3 Use the multiplication or division property of equality to make the coefficient of the

variable one.

Step 2

The next examples illustrate this step-by-step process for solving equations. Study these

examples carefully and be sure that you understand each step taken in the solution process.

Classroom Example

Solve 7m ϩ 1 Ϫ 3m ϭ 21.

Solve 5y Ϫ 4 ϩ 3y ϭ 12 .

EXAMPLE 1

Solution

5y Ϫ 4 ϩ 3y ϭ 12

8y Ϫ 4 ϭ 12

8y Ϫ 4 ϩ 4 ϭ 12 ϩ 4

8y ϭ 16

8y

16

ϭ

8

8

Combine similar terms on the left side

Divide both sides by 8

yϭ2

The solution set is 526. You can do the check alone now!

Classroom Example

Solve 8w ϩ 5 ϭ 2w Ϫ 3.

Solve 7x Ϫ 2 ϭ 3x ϩ 9.

EXAMPLE 2

Solution

Notice that both sides of the equation are in simplified form; thus we can begin by applying

the subtraction property of equality.

7x Ϫ 2 ϭ 3x ϩ 9

7x Ϫ 2 Ϫ 3x ϭ 3x ϩ 9 Ϫ 3x

Subtract 3x from both sides

4x Ϫ 2 ϭ 9

4x Ϫ 2 ϩ 2 ϭ 9 ϩ 2

4x ϭ 11

4x

11

ϭ

4

4

The solution set is e

Classroom Example

Solve 3d ϩ 1 ϭ 4d Ϫ 3.

EXAMPLE 3

Divide both sides by 4

11

4

11

f.

4

Solve 5n ϩ 12 ϭ 9n Ϫ 16.

Solution

5n ϩ 12 ϭ 9n Ϫ 16

5n ϩ 12 Ϫ 9n ϭ 9n Ϫ 16 Ϫ 9n

Subtract 9n from both sides

Ϫ4n ϩ 12 ϭ Ϫ16

Ϫ4n ϩ 12 Ϫ 12 ϭ Ϫ16 Ϫ 12

Subtract 12 from both sides

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