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3: More on Solving Equations and Problem Solving

# 3: More on Solving Equations and Problem Solving

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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

Add 4 to both sides

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

Add 2 to both sides

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

3.3 • More on Solving Equations and Problem Solving

105

Ϫ4n ϭ Ϫ28

Ϫ4n

Ϫ28

ϭ

Ϫ4

Ϫ4

Divide both sides by Ϫ4

nϭ7

The solution set is 576.

Classroom Example

Solve 2m ϩ 5 ϭ 2m Ϫ 3.

EXAMPLE 4

Solve 3x ϩ 8 ϭ 3x Ϫ 2.

Solution

3x ϩ 8 ϭ 3x Ϫ 2

3x ϩ 8 Ϫ 3x ϭ 3x Ϫ 2 Ϫ 3x

8 ϭ Ϫ2

Subtract 3x from both sides

False statement

Since we obtained an equivalent equation that is a false statement, there is no value of x that

will make the equation a true statement. When the equation is not true under any condition,

then the equation is called a contradiction. The solution set for an equation that is a contradiction is the empty or null set, and it is symbolized by л.

Classroom Example

Solve 3n ϩ 4 ϭ 8n ϩ 4 Ϫ 5n.

EXAMPLE 5

Solve 4x ϩ 6 Ϫ x ϭ 3x ϩ 6.

Solution

4x ϩ 6 Ϫ x ϭ 3x ϩ 6

3x ϩ 6 ϭ 3x ϩ 6

Combine similar terms on the left side

3x ϩ 6 Ϫ 3x ϭ 3x ϩ 6 Ϫ 3x

Subtract 3x from both sides

6ϭ6

True statement

Since we obtained an equivalent equation that is a true statement, any value of x will make

the equation a true statement. When an equation is true for any value of the variable, the equation is called an identity. The solution set for an equation that is an identity is the set of all

real numbers. We will denote the set of all real numbers as {all reals}.

Word Problems

As we expand our skills for solving equations, we also expand our capabilities for solving

word problems. No one definite procedure will ensure success at solving word problems, but

the following suggestions can be helpful.

Suggestions for Solving Word Problems

1. Read the problem carefully, and make sure that you understand the meanings of all

the words. Be especially alert for any technical terms in the statement of the

problem.

2. Read the problem a second time (perhaps even a third time) to get an overview of

the situation being described and to determine the known facts as well as what is to

be found.

3. Sketch any figure, diagram, or chart that might be helpful in analyzing the problem.

(continued)

106

Chapter 3 • Equations, Inequalities, and Problem Solving

4. Choose a meaningful variable to represent an unknown quantity in the problem

(perhaps t if time is an unknown quantity); represent any other unknowns in terms

of that variable.

5. Look for a guideline that you can use to set up an equation. A guideline might be a

formula such as distance equals rate times time or a statement of a relationship such

as the sum of the two numbers is 28. A guideline may also be indicated by a figure

or diagram that you sketch for a particular problem.

6. Form an equation that contains the variable and that translates the conditions of the

guideline from English into algebra.

7. Solve the equation and use the solution to determine all facts requested in the

problem.

8. Check all answers by going back to the original statement of the problem and

verifying that the answers make sense.

If you decide not to check an answer, at least use the reasonableness-of-answer idea as a partial check. That is to say, ask yourself the question: Is this answer reasonable? For example,

if the problem involves two investments that total \$10,000, then an answer of \$12,000 for one

investment is certainly not reasonable.

Now let’s consider some examples and use these suggestions as you work them out.

Consecutive Number Problems

Some problems involve consecutive numbers or consecutive even or odd numbers. For

instance, 7, 8, 9, and 10 are consecutive numbers. To solve these applications, you must know

how to represent consecutive numbers with variables. Let n represent the first number. For

consecutive numbers, the next number is 1 more and is represented by n ϩ 1. To continue,

we add 1 to each preceding expression, obtaining the representations shown here:

7

8

9

10

n

nϩ1

nϩ2

nϩ3

The pattern is somewhat different for consecutive even or odd numbers. For example, 2, 4, 6,

and 8 are consecutive even numbers. Let n represent the first even number; then n ϩ 2 represents the next even number. To continue, we add 2 to each preceding expression, obtaining

these representations:

2

4

6

8

n

nϩ2

nϩ4

nϩ6

Consecutive odd numbers have the same pattern of adding 2 to each preceding expression

because consecutive odd numbers are two odd numbers with one and only one whole number between them.

Classroom Example

Find two consecutive even numbers

whose sum is 42.

EXAMPLE 6

Find two consecutive even numbers whose sum is 74.

Solution

Let n represent the first number; then n ϩ 2 represents the next even number. Since their sum

is 74, we can set up and solve the following equation:

n ϩ 1n ϩ 22 ϭ 74

2n ϩ 2 ϭ 74

3.3 • More on Solving Equations and Problem Solving

107

2n ϩ 2 Ϫ 2 ϭ 74 Ϫ 2

2n ϭ 72

2n 72

ϭ

2

2

n ϭ 36

If n ϭ 36, then n ϩ 2 ϭ 38; thus, the numbers are 36 and 38.

✔ Check

To check your answers for Example 6, determine whether the numbers satisfy the conditions

stated in the original problem. Because 36 and 38 are two consecutive even numbers, and

36 ϩ 38 ϭ 74 (their sum is 74), we know that the answers are correct.

The fifth entry in our list of problem-solving suggestions is to look for a guideline that

can be used to set up an equation. The guideline may not be stated explicitly in the problem

but may be implied by the nature of the problem. Consider the following example.

Classroom Example

Tyron sells appliances on a salaryplus-commission basis. He receives

a monthly salary of \$550 and a

commission of \$95 for each appliance

that he sells. How many appliances

must he sell in a month to earn a total

monthly salary of \$1,120.

EXAMPLE 7

Barry sells bicycles on a salary-plus-commission basis. He receives a weekly salary of \$300

and a commission of \$15 for each bicycle that he sells. How many bicycles must he sell in a week

to earn a total weekly salary of \$750?

Solution

Let b represent the number of bicycles to be sold in a week. Then 15b represents his commission for those bicycles. The guideline “fixed salary plus commission equals total weekly

salary” generates the following equation:

Fixed salary ϩ Commission ϭ Total weekly salary

\$300

ϩ

15b

ϭ

\$750

Solving this equation yields

300 ϩ 15b Ϫ 300 ϭ 750 Ϫ 300

15b ϭ 450

15b 450

ϭ

15

15

b ϭ 30

Barry must sell 30 bicycles per week. (Does this number check?)

Geometric Problems

Sometimes the guideline for setting up an equation to solve a problem is based on a geometric relationship. Several basic geometric relationships pertain to angle measure. Let’s state

some of these relationships and then consider some examples.

1. Two angles for which the sum of their measure is 90Њ (the symbol Њ indicates degrees)

are called complementary angles.

2. Two angles for which the sum of their measure is 180Њ are called supplementary

angles.

3. The sum of the measures of the three angles of a triangle is 180Њ.

108

Chapter 3 • Equations, Inequalities, and Problem Solving

Classroom Example

One of two supplementary angles is

26° smaller than the other. Find the

measure of each of the angles.

EXAMPLE 8

One of two complementary angles is 14Њ larger than the other. Find the measure of each of

the angles.

Solution

If we let a represent the measure of the smaller angle, then a ϩ 14 represents the measure of

the larger angle. Since they are complementary angles, their sum is 90°, and we can proceed

as follows:

a ϩ 1a ϩ 142 ϭ 90

2a ϩ 14 ϭ 90

2a ϩ 14 Ϫ 14 ϭ 90 Ϫ 14

2a ϭ 76

2a 76

ϭ

2

2

a ϭ 38

If a ϭ 38, then a ϩ 14 ϭ 52, and the angles measure 38Њ and 52Њ.

Classroom Example

Find the measures of the three angles

of a triangle if the second angle is

twice the first angle, and the third

angle is half the second angle.

EXAMPLE 9

Find the measures of the three angles of a triangle if the second is three times the first and

the third is twice the second.

Solution

If we let a represent the measure of the smallest angle, then 3a and 2(3a) represent the measures of the other two angles. Therefore, we can set up and solve the following equation:

a ϩ 3a ϩ 213a2 ϭ 180

a ϩ 3a ϩ 6a ϭ 180

10a ϭ 180

10a 180

ϭ

10

10

a ϭ 18

If a ϭ 18, then 3a ϭ 54 and 213a2 ϭ 108. So the angles have measures of 18Њ, 54Њ, and 108Њ.

Concept Quiz 3.3

For Problems 1–10, answer true or false.

1. If n represents a whole number, then n ϩ 1 would represent the next consecutive whole

number.

2. If n represents an odd whole number, then n ϩ 1 would represent the next consecutive

odd whole number.

3. If n represents an even whole number, then n ϩ 2 would represent the next consecutive

even whole number.

4. The sum of the measures of two complementary angles is 90Њ.

5. The sum of the measures of two supplementary angles is 360Њ.

6. The sum of the measures of the three angles of a triangle is 120Њ.

7. When checking word problems, it is sufficient to check the solution in the equation.

8. For a word problem, the reasonableness of an answer is appropriate as a partial check.

9. For a conditional equation, the solution set is the set of all real numbers.

10. The solution set for an equation that is a contradiction is the null set.

3.3 • More on Solving Equations and Problem Solving

109

Problem Set 3.3

For Problems 1– 32, solve each equation. (Objectives 1–3)

1. 2x ϩ 7 ϩ 3x ϭ 32

2. 3x ϩ 9 ϩ 4x ϭ 30

30. 4x Ϫ 3 ϩ 2x ϭ 8x Ϫ 3 Ϫ x

31. Ϫ7 Ϫ 2n Ϫ 6n ϭ 7n Ϫ 5n ϩ 12

32. Ϫ3n ϩ 6 ϩ 5n ϭ 7n Ϫ 8n Ϫ 9

3. 7x Ϫ 4 Ϫ 3x ϭ Ϫ36

4. 8x Ϫ 3 Ϫ 2x ϭ Ϫ45

5. 3y Ϫ 1 ϩ 2y Ϫ 3 ϭ 4

6. y ϩ 3 ϩ 2y Ϫ 4 ϭ 6

Solve each word problem by setting up and solving an algebraic equation. (Objectives 4 and 5)

33. The sum of a number plus four times the number is 85.

What is the number?

7. 5n Ϫ 2 Ϫ 8n ϭ 31

34. A number subtracted from three times the number

yields 68. Find the number.

8. 6n Ϫ 1 Ϫ 10n ϭ 51

35. Find two consecutive odd numbers whose sum is 72.

9. Ϫ2n ϩ 1 Ϫ 3n ϩ n Ϫ 4 ϭ 7

10. Ϫn ϩ 7 Ϫ 2n ϩ 5n Ϫ 3 ϭ Ϫ6

11. 3x ϩ 4 ϭ 2x Ϫ 5

36. Find two consecutive even numbers whose sum is 94.

37. Find three consecutive even numbers whose sum is 114.

12. 5x Ϫ 2 ϭ 4x ϩ 6

13. 5x Ϫ 7 ϭ 6x Ϫ 9

14. 7x Ϫ 3 ϭ 8x Ϫ 13

15. 6x ϩ 1 ϭ 3x Ϫ 8

16. 4x Ϫ 10 ϭ x ϩ 17

17. 7y Ϫ 3 ϭ 5y ϩ 10

18. 8y ϩ 4 ϭ 5y Ϫ 4

19. 8n Ϫ 2 ϭ 8n Ϫ 7

20. 7n Ϫ 10 ϭ 9n Ϫ 13

21. Ϫ2x Ϫ 7 ϭ Ϫ3x ϩ 10

38. Find three consecutive odd numbers whose sum is 159.

39. Two more than three times a certain number is the same

as 4 less than seven times the number. Find the number.

40. One more than five times a certain number is equal to

eleven less than nine times the number. What is the

number?

41. The sum of a number and five times the number equals

eighteen less than three times the number. Find the

number.

42. One of two supplementary angles is five times as large

as the other. Find the measure of each angle.

22. Ϫ4x ϩ 6 ϭ Ϫ5x Ϫ 9

23. Ϫ3x ϩ 5 ϭ Ϫ5x Ϫ 8

24. Ϫ4x ϩ 7 ϭ Ϫ4x ϩ 4

25. Ϫ7 Ϫ 6x ϭ 9 Ϫ 9x

26. Ϫ10 Ϫ 7x ϭ 14 Ϫ 12x

27. 2x Ϫ 1 Ϫ x ϭ x Ϫ 1

28. 3x Ϫ 4 Ϫ 4x ϭ Ϫ5x ϩ 4x Ϫ 4

29. 5n Ϫ 4 Ϫ n ϭ Ϫ3n Ϫ 6 ϩ n

43. One of two complementary angles is 6Њ smaller than

twice the other angle. Find the measure of each angle.

44. If two angles are complementary and the difference

between their measures is 62Њ, find the measure of each

angle.

45. If two angles are supplementary and the larger angle is

20Њ less than three times the smaller angle, find the measure of each angle.

46. Find the measures of the three angles of a triangle if the

largest is 14Њ less than three times the smallest, and the

other angle is 4Њ larger than the smallest.

110

Chapter 3 • Equations, Inequalities, and Problem Solving

47. One of the angles of a triangle has a measure of 40Њ.

Find the measures of the other two angles if the difference between their measures is 10Њ.

The other side is 10 yards longer than the shortest side.

Find the lengths of the three sides of the lot.

48. Jesstan worked as a telemarketer on a salary-pluscommission basis. He was paid a salary of \$300 a week

and a \$12 commission for each sale. If his earnings for

the week were \$960, how many sales did he make?

49. Marci sold an antique vase in an online auction for

\$69.00. This was \$15 less than twice the amount she

paid for it. What price did she pay for the vase?

50. A set of wheels sold in an online auction for \$560. This

was \$35 more than three times the opening bid. How

much was the opening bid?

51. Suppose that Bob is paid two times his normal hourly

rate for each hour he works over 40 hours in a week.

Last week he earned \$504 for 48 hours of work. What

is his hourly wage?

52. Last week on an algebra test, the highest grade was 9

points less than three times the lowest grade. The sum

of the two grades was 135. Find the lowest and highest

grades on the test.

53. At a university-sponsored concert, there were three

times as many women as men. A total of 600 people

attended the concert. How many men and how many

women attended?

54. Suppose that a triangular lot is enclosed with 135 yards

of fencing (see Figure 3.1). The longest side of the lot is

5 yards longer than twice the length of the shortest side.

Figure 3.1

55. The textbook for a biology class cost \$15 more than

twice the cost of a used textbook for college algebra. If

the cost of the two books together is \$129, find the cost

of the biology book.

56. A nutrition plan counts grams of fat, carbohydrates, and

fiber. The grams of carbohydrates are to be 15 more than

twice the grams of fat. The grams of fiber are to be three

less than the grams of fat. If the grams of carbohydrate,

fat, and fiber must total 48 grams for a dinner meal, how

many grams of each would be in the meal?

57. At a local restaurant, \$275 in tips is to be shared between

the server, bartender, and busboy. The server gets \$25

more than three times the amount the busboy receives.

The bartender gets \$50 more than the amount the busboy

receives. How much will the server receive?

Additional word problems can be found in Appendix B.

All the problems in the Appendix marked as (3.3) are

appropriate for this section.

Thoughts Into Words

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

the equation 3x ϩ 4 ϭ 5x Ϫ 2.

59. Suppose your friend solved the problem find two consecutive odd integers whose sum is 28 as follows:

x ϩ 1x ϩ 12 ϭ 28

2x ϭ 27

27

1

ϭ 13

2

2

1

She claims that 13 will check in the equation. Where

2

has she gone wrong, and how would you help her?

Further Investigations

60. Solve each of these equations.

(a) 7x Ϫ 3 ϭ 4x Ϫ 3

(b) Ϫx Ϫ 4 ϩ 3x ϭ 2x Ϫ 7

(c) Ϫ3x ϩ 9 Ϫ 2x ϭ Ϫ5x ϩ 9

(d) 5x Ϫ 3 ϭ 6x Ϫ 7 Ϫ x

(e) 7x ϩ 4 ϭ Ϫx ϩ 4 ϩ 8x

(f) 3x Ϫ 2 Ϫ 5x ϭ 7x Ϫ 2 Ϫ 5x

(g) Ϫ6x Ϫ 8 ϭ 6x ϩ 4

(h) Ϫ8x ϩ 9 ϭ Ϫ8x ϩ 5

61. Make up an equation whose solution set is the null set and

explain why.

62. Make up an equation whose solution set is the set of all real

numbers and explain why.

3.4 • Equations Involving Parentheses and Fractional Forms

Answers to the Concept Quiz

1. True

2. False

3. True

9. False

10. True

3.4

4. True

5. False

6. False

7. False

111

8. True

Equations Involving Parentheses and Fractional Forms

OBJECTIVES

1

Solve ﬁrst-degree equations that involve the use of the distributive

property

2

Solve ﬁrst-degree equations that involve fractional forms

3

Solve a variety of word problems involving ﬁrst-degree equations

We will use the distributive property frequently in this section as we add to our techniques

for solving equations. Recall that in symbolic form the distributive property states that

a(b ϩ c) ϭ ab ϩ ac. Consider the following examples, which illustrate the use of this property to remove parentheses. Pay special attention to the last two examples, which involve a

negative number in front of the parentheses.

иxϩ3и2

5 и y Ϫ 5 и 3

3(x ϩ 2) ϭ

3

5(y Ϫ 3) ϭ

2(4x ϩ 7) ϭ

ϭ 3x ϩ 6

ϭ 5y Ϫ 15

a(b Ϫ c) ϭ ab Ϫ ac

2(4x) ϩ 2(7)    ϭ 8x ϩ 14

Ϫ1(n ϩ 4) ϭ   (Ϫ1)(n) ϩ (Ϫ1)(4)    ϭ Ϫn Ϫ 4

Ϫ6(x Ϫ 2) ϭ   (Ϫ6)(x) Ϫ (Ϫ6)(2)    ϭ Ϫ6x ϩ 12

Do this step mentally!

It is often necessary to solve equations in which the variable is part of an expression enclosed

in parentheses. We use the distributive property to remove the parentheses, and then we proceed in the usual way. Consider the next examples. (Notice that we are beginning to show

only the major steps when solving an equation.)

Classroom Example

Solve 2(d Ϫ 5) ϭ 5.

Solve 31x ϩ 22 ϭ 23.

EXAMPLE 1

Solution

31x ϩ 22 ϭ 23

3x ϩ 6 ϭ 23

Applied distributive property to left side

3x ϭ 17

Subtracted 6 from both sides

17

3

The solution set is e

Divided both sides by 3

17

f.

3

112

Chapter 3 • Equations, Inequalities, and Problem Solving

Classroom Example

Solve 3(m Ϫ 1) ϭ 6 (m ϩ 2).

EXAMPLE 2

Solve 41x ϩ 32 ϭ 21x Ϫ 62 .

Solution

41x ϩ 32 ϭ 21x Ϫ 62

4x ϩ 12 ϭ 2x Ϫ 12

2x ϩ 12 ϭ Ϫ12

2x ϭ Ϫ24

x ϭ Ϫ12

Applied distributive property on each side

Subtracted 2x from both sides

Subtracted 12 from both sides

Divided both sides by 2

The solution set is 5Ϫ126.

It may be necessary to remove more than one set of parentheses and then to use the

distributive property again to combine similar terms. Consider the following two examples.

Classroom Example

Solve 2(t ϩ 4) ϩ 6(t ϩ 5) ϭ 32.

EXAMPLE 3

Solve 51w ϩ 32 ϩ 31w ϩ 12 ϭ 14.

Solution

51w ϩ 32 ϩ 31w ϩ 12 ϭ 14

5w ϩ 15 ϩ 3w ϩ 3 ϭ 14

8w ϩ 18 ϭ 14

8w ϭ Ϫ4

4

8

1

wϭϪ

2

wϭϪ

Applied distributive property

Combined similar terms

Subtracted 18 from both sides

Divided both sides by 8

Reduced

1

The solution set is eϪ f .

2

Classroom Example

Solve 4(w Ϫ 3) Ϫ 8(w Ϫ 2) ϭ 9.

EXAMPLE 4

Solve 61x Ϫ 72 Ϫ 21x Ϫ 42 ϭ 13.

Solution

61x Ϫ 72 Ϫ 21x Ϫ 42 ϭ 13

6x Ϫ 42 Ϫ 2x ϩ 8 ϭ 13

4x Ϫ 34 ϭ 13

4x ϭ 47

47

4

The solution set is e

47

f.

4

Be careful with this sign!

Distributive property

Combined similar terms

Added 34 to both sides

Divided both sides by 4

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3: More on Solving Equations and Problem Solving

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