6.3 Applications of Linear Equations - Moreland MATH Algebra: Equations and Inequalities > 6.3 Applications of Linear Equations > Solving a Formula for One of Its Variables `

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6 Algebra: Equations and Inequalities > 6.3 Applications of Linear Equations

6.3 Applications of Linear Equations

What am I Supposed to Learn?After you have read this section, you should be able to:

1 Use linear equations to solve problems.

2 Solve a formula for a variable.

How Long It Takes to Earn $1000

dSource: Time

In this section, you'll see examples and exercises focused on how much money Americans earn. These situations illustrate a step-by-step strategy for solving problems.As you become familiar with this strategy, you will learn to solve a wide variety of problems.

Problem Solving with Linear Equations

1 Use linear equations to solve problems.

We have seen that a model is a mathematical representation of a real-world situation. In this section, we will be solving problems that are presented in English. Thismeans that we must obtain models by translating from the ordinary language of English into the language of algebraic equations. To translate, however, we mustunderstand the English prose and be familiar with the forms of algebraic language. Following are some general steps we will follow in solving word problems.

Strategy for Solving Word ProblemsStep 1 Read the problem carefully several times until you can state in your own words what is given and what the problem is looking for. Let x (or any variable)represent one of the unknown quantities in the problem.

Step 2 If necessary, write expressions for any other unknown quantities in the problem in terms of x.

Step 3 Write an equation in x that models the verbal conditions of the problem.

Step 4 Solve the equation and answer the problem's question.

Step 5 Check the solution in the original wording of the problem, not in the equation obtained from the words.

The most difficult step in this process is step 3 because it involves translating verbal conditions into an algebraic equation. Translations of some commonly used Englishphrases are listed in Table 6.2 on the next page. We choose to use x to represent the variable, but we can use any letter.

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TABLE 6.2 Algebraic Translations of English PhrasesEnglish Phrase Algebraic Expression

Addition

The sum of a number and 7

Five more than a number; a number plus 5

A number increased by 6; 6 added to a number

Subtraction A number minus 4

A number decreased by 5

A number subtracted from 8

The difference between a number and 6

The difference between 6 and a number

Seven less than a number

Seven minus a number

Nine fewer than a number

Multiplication

Five times a number

The product of 3 and a number

Two-thirds of a number (used with fractions)

Seventy-five percent of a number (used with decimals)

Thirteen multiplied by a number

A number multiplied by 13

Twice a number

5x

3x

0.75x

13x

13x

2x

Division

A number divided by 3

The quotient of 7 and a number

The quotient of a number and 7

The reciprocal of a number

More than one operation

The sum of twice a number and 7

Twice the sum of a number and 7

Three times the sum of 1 and twice a number

Nine subtracted from 8 times a number

Twenty-five percent of the sum of 3 times a number and 14

Seven times a number, increased by 24

Seven times the sum of a number and 24

Great Question!Table 6.2 looks long and intimidating. What's the best way to get through the table?

Cover the right column with a sheet of paper and attempt to formulate the algebraic expression for the English phrase in the left column on your own. Then slide thepaper down and check your answer. Work through the entire table in this manner.



x + 7

x + 5

x + 6

x − 4

x − 5

8 − x

x − 6

6 − x

x − 7

7 − x

x − 9

x23

x

3

7x

x

7

1x

2x + 7

2 (x + 7)

3 (1 + 2x)

8x − 9

0.25 (3x + 14)

7x + 24

7 (x + 24)







Page 366

Example 1 Starting Salaries for College Graduates with Undergraduate DegreesThe bar graph in Figure 6.4 shows the ten most popular college majors with median, or middlemost, starting salaries for recent college graduates.

dFIGURE 6.4 Source: PayScale (2010 data)









Page 367


The median starting salary of a business major exceeds that of a psychology major by $8 thousand. The median starting salary of an English major exceeds that of apsychology major by $3 thousand. Combined, their median starting salaries are $116 thousand. Determine the median starting salaries of psychology majors,business majors, and English majors with bachelor's degrees.

SOLUTION

Step 1 Let x represent one of the unknown quantities. We know something about the median starting salaries of business majors and English majors: Businessmajors earn $8 thousand more than psychology majors and English majors earn $3 thousand more than psychology majors. We will let

Step 2 Represent other unknown quantities in terms of x. Because business majors earn $8 thousand more than psychology majors, let

Because English majors earn $3 thousand more than psychology majors, let

Step 3 Write an equation in x that models the conditions. Combined, the median starting salaries for psychology, business, and English majors are $116thousand.

Step 4 Solve the equation and answer the question.

Thus,

The median starting salary of psychology majors is $35 thousand, the median starting salary of business majors is $43 thousand, and the median starting salary ofEnglish majors is $38 thousand.

Step 5 Check the proposed solution in the original wording of the problem. The problem states that combined, the median starting salaries are $116thousand. Using the median salaries we determined in step 4, the sum is

or $116 thousand, which verifies the problem's conditions.




x = the median starting salary, in thousands of dollars, of psychology majors.

►x + 8 = the median starting salary, in thousands of dollars, of business majors.

Salary exceeds

a psychology

major, x, by

$8 thousand.

►x + 3 = the median starting salary, in thousands of dollars, of English majors.

Salary exceeds

a psychology

major, x, by

$3 thousand.

The median

starting salary

for psychology

majors

▼

x

plus

▼

+

the median

starting salary

for business

majors

▼

(x + 8)

plus

▼

+

the median

starting salary

for English

majors

▼

(x + 3)

is

▼

=

$116 thousand.

▼

116

x + (x + 8) + (x + 3)

3x + 11

3x

x

= 116

= 116

= 105

= 35

This is the equation that models the problem's conditions.

Remove parentheses, regroup, and combine like terms.

Subtract 11 from both sides.

Divide both sides by 3.

starting salary of psychology majors

starting salary of business majors

starting salary of English majors

= x = 35

= x + 8 = 35 + 8 = 43

= x + 3 = 35 + 3 = 38.

$35 thousand + $43 thousand + $38 thousand,













Great Question!Example 1 involves using the word exceeds to represent two of the unknown quantities. Can you help me to write algebraic expressions for quantitiesdescribed using exceeds?

Modeling with the word exceeds can be a bit tricky. It's helpful to identify the smaller quantity. Then add to this quantity to represent the larger quantity. For example,suppose that Tim's height exceeds Tom's height by a inches. Tom is the shorter person. If Tom's height is represented by x, then Tim's height is represented by

Check Point 1Three of the bars in Figure 6.4 on page 366 represent median starting salaries of education, computer science, and economics majors. The median starting salary ofa computer science major exceeds that of an education major by $21 thousand. The median starting salary of an economics major exceeds that of an education majorby $14 thousand. Combined, their median starting salaries are $140 thousand. Determine the median starting salaries of education majors, computer science majors,and economics majors with bachelor's degrees.

Example 2 Modeling Attitudes of College Freshmen

Researchers have surveyed college freshmen every year since 1969. Figure 6.5 shows that attitudes about some life goals have changed dramatically over theyears. In particular, the freshman class of 2010 was more interested in making money than the freshmen of 1969 had been. In 1969, 42% of first-year college studentsconsidered “being well-off financially” essential or very important. For the period from 1969 through 2010, this percentage increased by approximately 0.9 each year. Ifthis trend continues, by which year will all college freshmen consider “being well-off financially” essential or very important?

dFIGURE 6.5 Source: Higher Education Research Institute

SOLUTION



x + a.








Page 368

Step 1 Let x represent one of the unknown quantities. We are interested in the year when all college freshmen, or 100% of the freshmen, will consider this lifeobjective essential or very important. Let

Step 2 Represent other unknown quantities in terms of x. There are no other unknown quantities to find, so we can skip this step.


x = the number of years after 1969 when all freshmen will consider “being well-off financially” essential or very important.








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Step 3 Write an equation in x that models the conditions.


Using current trends, by approximately 64 years after 1969, or in 2033, all freshmen will consider “being well-off financially” essential or very important.

Step 5 Check the proposed solution in the original wording of the problem. The problem states that all freshmen (100%, represented by 100 using the model)will consider the objective essential or very important. Does this approximately occur if we increase the 1969 percentage, 42%, by 0.9 each year for 64 years, ourproposed solution?

This verifies that using trends shown in Figure 6.5, all first-year college students will consider the objective essential or very important approximately 64 years after1969.

Check Point 2Figure 6.5 shows that the freshman class of 2010 was less interested in developing a philosophy of life than the freshmen of 1969 had been. In 1969, 85% of thefreshmen considered this objective essential or very important. Since then, this percentage has decreased by approximately 0.9 each year. If this trend continues, bywhich year will only 25% of college freshmen consider “developing a meaningful philosophy of life” essential or very important?

A Brief Review: Clearing an Equation of Decimals• You can clear an equation of decimals by multiplying each side by a power of 10. The exponent on 10 will be equal to the greatest number of digits to the right ofany decimal point in the equation.

• Multiplying a decimal number by has the effect of moving the decimal point places to the right.

Example

The greatest number of digits to the right of any decimal point in the equation is 1. Multiply each side by or 10.

It is not a requirement to clear decimals before solving an equation. Compare this solution to the one in step 4 of Example 2. Which method do you prefer?




The 1969

percentage

▼

42

increased

by

▼

+

0.9 each year

for x years

▼

0.9x

equals

▼

=

100% of the

freshmen.

▼

100

42 + 0.9x

42 − 42 + 0.9x

0.9x

0.9x

0.9

x

= 100

= 100 − 42

= 58

= 58

0.9

= 64. ≈ 644̄



Simplify.

Divide both sides by 0.9.

Simplify and round to the nearest whole number.

42 + 0.9 (64) = 42 + 57.6 = 99.6 ≈ 100

10nn

42 + 0.9x = 100

,101

10(42 + 0.9x)

10(42) + 10(0.9x)

420 + 9x

420 − 420 + 9x

9x

9x

x

= 10(100)

= 10(100)

= 100

= 1000 − 420

= 580

= 580

9

= 64. ≈ 644̄












Page 370


Great Question!Why are algebraic word problems important?

There is great value in reasoning through the steps for solving a word problem. This value comes from the problem-solving skills that you will attain and is often moreimportant than the specific problem or its solution.

Example 3 Selecting a Monthly Text Message PlanYou are choosing between two texting plans. Plan A has a monthly fee of $20.00 with a charge of $0.05 per text. Plan B has a monthly fee of $5.00 with a charge of$0.10 per text. Both plans include photo and video texts. For how many text messages will the costs for the two plans be the same?

SOLUTION

Step 1 Let x represent one of the unknown quantities. Let

Step 2 Represent other unknown quantities in terms of x. There are no other unknown quantities, so we can skip this step.

Step 3 Write an equation in x that models the conditions. The monthly cost for plan A is the monthly fee, $20.00, plus the per-text charge, $0.05, times thenumber of text messages, x. The monthly cost for plan B is the monthly fee, $5.00, plus the per-text charge, $0.10, times the number of text messages, x.


Because x represents the number of text messages for which the two plans cost the same, the costs will be the same for 300 texts per month.

Step 5 Check the proposed solution in the original wording of the problem. The problem states that the costs for the two plans should be the same. Let's seeif they are with 300 text messages:

With 300 text messages, both plans cost $35 for the month. Thus, the proposed solution, 300 text messages, satisfies the problem's conditions.

Check Point 3You are choosing between two texting plans. Plan A has a monthly fee of $15.00 with a charge of $0.08 per text. Plan B has a monthly fee of $3.00 with a charge of$0.12 per text. For how many text messages will the costs for the two plans be the same?




x = the number of text messages for which the two plans cost the same.

The monthly

cost for plan A

▼

20 + 0.05x

must

equal

▼

=

the monthly

cost for plan B.

▼

5 + 0.10x

20 + 0.05x

20

1515

0.05

300

= 5 + 0.10x

= 5 + 0.05x

= 0.05x

= 0.05x

0.05

= x


Subtract 0.05x from both sides.


Divide both sides by 0.05.

Simplify.

Cost for plan A = $20 + $0.05 (300) = $20 + $15 = $35

▲ ▲

Monthly fee Per-text charge

▼ ▼

Cost for plan B = $5 + $0.10 (300) = $5 + $30 = $35.












6 Algebra: Equations and Inequalities > 6.3 Applications of Linear Equations > Solving a Formula for One of Its Variables

Example 4 A Price Reduction on a Digital CameraYour local computer store is having a terrific sale on digital cameras. After a 40% price reduction, you purchase a digital camera for $276. What was the camera'sprice before the reduction?

SOLUTION

Step 1 Let x represent one of the unknown quantities. We will let

Step 2 Represent other unknown quantities in terms of x. There are no other unknown quantities to find, so we can skip this step.

Step 3 Write an equation in x that models the conditions. The camera's original price minus the 40% reduction is the reduced price, $276.


d

The digital camera's price before the reduction was $460.

Step 5 Check the proposed solution in the original wording of the problem. The price before the reduction, $460, minus the 40% reduction should equal thereduced price given in the original wording, $276:

This verifies that the digital camera's price before the reduction was $460.

Check Point 4After a 30% price reduction, you purchase a new computer for $840. What was the computer's price before the reduction?

Great Question!Why is the 40% reduction written as 0.4x in Example 4?

• 40% is written 0.40 or 0.4.

• “Of” represents multiplication, so 40% of the original price is

Notice that the original price, x, reduced by 40% is and not

Solving a Formula for One of Its Variables

2 Solve a formula for a variable.

We know that solving an equation is the process of finding the number (or numbers) that make the equation a true statement. All of the equations we have solvedcontained only one letter, x.

By contrast, formulas contain two or more letters, representing two or more variables. An example is the formula for the perimeter of a rectangle:



x = the original price of the digital camera prior to the reduction.

Original

price

▼

x

minus

▼

−

the reduction

(40% of the

original price)

▼

0.4x

is

▼

=

the reduced

price, $276.

▼

276

460 − 40% of 460 = 460 − 0.4 (460) = 460 − 184 = 276.

0.4x.

x − 0.4x x − 0.4.

P = 2l + 2w. ◄A rectangle's perimeter is

the sum of twice its length

and twice its width.








Page 371

We say that this formula is solved for the variable P because P is alone on one side of the equation and the other side does not contain a P.

Solving a formula for a variable means rewriting the formula so that the variable is isolated on one side of the equation. It does not mean obtaining a numerical valuefor that variable.









Page 372

6 Algebra: Equations and Inequalities > 6.3 Applications of Linear Equations > Solving a Formula for One of Its Variables

To solve a formula for one of its variables, treat that variable as if it were the only variable in the equation. Think of the other variables as if they were numbers. Isolate allterms with the specified variable on one side of the equation and all terms without the specified variable on the other side. Then divide both sides by the same nonzeroquantity to get the specified variable alone. The next two examples show how to do this.

Example 5 Solving a Formula for a VariableSolve the formula for l.

SOLUTION

First, isolate on the right by subtracting from both sides. Then solve for l by dividing both sides by 2.

Equivalently,

Check Point 5Solve the formula for

Example 6 Solving a Formula for a VariableThe total price of an article purchased on a monthly deferred payment plan is described by the following formula:

In this formula, T is the total price, D is the down payment, p is the monthly payment, and m is the number of months one pays. Solve the formula for p.

SOLUTION

First, isolate pm on the right by subtracting D from both sides. Then, isolate p from pm by dividing both sides of the formula by m.

Check Point 6Solve the formula for m.




P = 2l + 2w

2l 2w

P

P − 2w

P − 2w

P−2w

2

P−2w

2

We need to isolate l.

▼

= 2l + 2w

= 2l + 2w − 2w

= 2l

=2l

2

= l

This is the given formula.

Isolate 2l by subtracting 2w from both sides.

Simplify.

Solve for l by dividing both sides by 2.

Simplify.

l = .P−2w

2

P = 2l + 2w w.

T = D + pm.

T

T − D

T − D

T−D

m

T−D

m

We need to

isolate p.

▼

= D + pm

= D − D + pm

= pm

=pm

m

= p

This is the given formula. We want p alone.

Isolate pm by subtracting D from both sides.

Simplify.

Now isolate p by dividing both sides by m.

Simplify : = = = p.pm

m

p m

m

p

1

T = D + pm













6 Algebra: Equations and Inequalities > 6.3 Applications of Linear Equations > Concept and Vocabulary Check

Concept and Vocabulary CheckFill in each blank so that the resulting statement is true.

1. According to the U.S. Office of Management and Budget, the 2011 budget for defense exceeded the budget for education by $658.6 billion. If x represents thebudget for education, in billions of dollars, the budget for defense can be represented by ___________________.

2. In 2000, 31% of U.S. adults viewed a college education as essential for success. For the period from 2000 through 2010, this percentage increased byapproximately 2.4 each year. The percentage of U.S. adults who viewed a college education as essential for success x years after 2000 can be represented by___________________.

3. A text message plan costs $4.00 per month plus $0.15 per text. The monthly cost for x text messages can be represented by ___________________.

4. I purchased a computer after a 15% price reduction. If x represents the computer's original price, the reduced price can be represented by________________________.

5. Solving a formula for a variable means rewriting the formula so that the variable is _________________________.

6. In order to solve for we first ___________________ and then ____________________.

Exercise Set 6.3Practice ExercisesUse the five-step strategy for solving word problems to find the number or numbers described in Exercises 1–10.

1. When five times a number is decreased by 4, the result is 26. What is the number?

2. When two times a number is decreased by 3, the result is 11. What is the number?

3. When a number is decreased by 20% of itself, the result is 20. What is the number?

4. When a number is decreased by 30% of itself, the result is 28. What is the number?

5. When 60% of a number is added to the number, the result is 192. What is the number?

6. When 80% of a number is added to the number, the result is 252. What is the number?

7. 70% of what number is 224?

8. 70% of what number is 252?

9. One number exceeds another by 26. The sum of the numbers is 64. What are the numbers?

10. One number exceeds another by 24. The sum of the numbers is 58. What are the numbers?

Practice PlusIn Exercises 11–18, write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number.

11. A number decreased by the sum of the number and four

12. A number decreased by the difference between eight and the number

13. Six times the product of negative five and a number

14. Ten times the product of negative four and a number

15. The difference between the product of five and a number and twice the number

16. The difference between the product of six and a number and negative two times the number

17. The difference between eight times a number and six more than three times the number

18. Eight decreased by three times the sum of a number and six

Application ExercisesHow will you spend your average life expectancy of 78 years? The bar graph shows the average number of years you will devote to each of your most time-consumingactivities. Exercises 19–20 are based on the data displayed by the graph.



y = mx + b x,








Page 373

dSource: U.S. Bureau of Labor Statistics

19. According to the U.S. Bureau of Labor Statistics, you will devote 37 years to sleeping and watching TV. The number of years sleeping will exceed the number ofyears watching TV by 19. Over your lifetime, how many years will you spend on each of these activities?

20. According to the U.S. Bureau of Labor Statistics, you will devote 32 years to sleeping and eating. The number of years sleeping will exceed the number of yearseating by 24. Over your lifetime, how many years will you spend on each of these activities?










The bar graph shows average yearly earnings in the United States for people with a college education, by final degree earned. Exercises 21–22 are based on the datadisplayed by the graph.

dSource: U.S. Census Bureau

21. The average yearly salary of an American whose final degree is a master's is $49 thousand less than twice that of an American whose final degree is a bachelor's.Combined, two people with each of these educational attainments earn $116 thousand. Find the average yearly salary of Americans with each of these final degrees.

22. The average yearly salary of an American whose final degree is a doctorate is $39 thousand less than twice that of an American whose final degree is abachelor's. Combined, two people with each of these educational attainments earn $126 thousand. Find the average yearly salary of Americans with each of thesefinal degrees.

Even as Americans increasingly view a college education as essential for success, many believe that a college education is becoming less available to qualifiedstudents. Exercises 23–24 are based on the data displayed by the graph.

dSource: Public Agenda

23. In 2000, 31% of U.S. adults viewed a college education as essential for success. For the period 2000 through 2010, the percentage viewing a college educationas essential for success increased on average by approximately 2.4 each year. If this trend continues, by which year will 67% of all American adults view a collegeeducation as essential for success?

24. The data displayed by the graph at the bottom of the previous column indicate that in 2000, 45% of U.S. adults believed most qualified students get to attendcollege. For the period from 2000 through 2010, the percentage who believed that a college education is available to most qualified students decreased byapproximately 1.7 each year. If this trend continues, by which year will only 11% of all American adults believe that most qualified students get to attend college?

On average, every minute of every day, 158 babies are born. The bar graph represents the results of a single day of births, deaths, and population increaseworldwide. Exercises 25–26 are based on the information displayed by the graph.










Page 374

dSource: James Henslin, Sociology, Eleventh Edition, Pearson, 2012.

25. Each day, the number of births in the world is 84 thousand less than three times the number of deaths.

a. If the population increase in a single day is 228 thousand, determine the number of births and deaths per day.

b. If the population increase in a single day is 228 thousand, by how many millions of people does the worldwide population increase each year? Round to thenearest million.

c. Based on your answer to part (b), approximately how many years does it take for the population of the world to increase by an amount greater than the entireU.S. population (315 million)?

26. Each day, the number of births in the world exceeds twice the number of deaths by 72 thousand.

a. If the population increase in a single day is 228 thousand, determine the number of births and deaths per day.

b. If the population increase in a single day is 228 thousand, by how many millions of people does the worldwide population increase each year? Round to thenearest million.

c. Based on your answer to part (b), approximately how many years does it take for the population of the world to increase by an amount greater than the entireU.S. population (315 million)?

27. A new car worth $24,000 is depreciating in value by $3000 per year. After how many years will the car's value be $9000?

28. A new car worth $45,000 is depreciating in value by $5000 per year. After how many years will the car's value be $10,000?










29. You are choosing between two health clubs. Club A offers membership for a fee of $40 plus a monthly fee of $25. Club B offers membership for a fee of $15 plus amonthly fee of $30. After how many months will the total cost at each health club be the same? What will be the total cost for each club?

30. You need to rent a rug cleaner. Company A will rent the machine you need for $22 plus $6 per hour. Company B will rent the same machine for $28 plus $4 perhour. After how many hours of use will the total amount spent at each company be the same? What will be the total amount spent at each company?

31. The bus fare in a city is $1.25. People who use the bus have the option of purchasing a monthly discount pass for $15.00. With the discount pass, the fare isreduced to $0.75. Determine the number of times in a month the bus must be used so that the total monthly cost without the discount pass is the same as the totalmonthly cost with the discount pass.

32. A discount pass for a bridge costs $30.00 per month. The toll for the bridge is normally $5.00, but it is reduced to $3.50 for people who have purchased thediscount pass. Determine the number of times in a month the bridge must be crossed so that the total monthly cost without the discount pass is the same as the totalmonthly cost with the discount pass.

33. You are choosing between two plans at a discount warehouse. Plan A offers an annual membership fee of $100 and you pay 80% of the manufacturer'srecommended list price. Plan B offers an annual membership fee of $40 and you pay 90% of the manufacturer's recommended list price. How many dollars ofmerchandise would you have to purchase in a year to pay the same amount under both plans? What will be the cost for each plan?

34. You are choosing between two plans at a discount warehouse. Plan A offers an annual membership fee of $300 and you pay 70% of the manufacturer'srecommended list price. Plan B offers an annual membership fee of $40 and you pay 90% of the manufacturer's recommended list price. How many dollars ofmerchandise would you have to purchase in a year to pay the same amount under both plans? What will be the cost for each plan?

35. In 2010, there were 13,300 students at college A, with a projected enrollment increase of 1000 students per year. In the same year, there were 26,800 students atcollege B, with a projected enrollment decline of 500 students per year. According to these projections, when will the colleges have the same enrollment? What will bethe enrollment in each college at that time?

36. In 2000, the population of Greece was 10,600,000, with projections of a population decrease of 28,000 people per year. In the same year, the population ofBelgium was 10,200,000, with projections of a population decrease of 12,000 people per year. (Source: United Nations) According to these projections, when will thetwo countries have the same population? What will be the population at that time?

37. After a 20% reduction, you purchase a television for $336. What was the television's price before the reduction?

38. After a 30% reduction, you purchase a dictionary for $30.80. What was the dictionary's price before the reduction?

39. Including 8% sales tax, an inn charges $162 per night. Find the inn's nightly cost before the tax is added.

40. Including 5% sales tax, an inn charges $252 per night. Find the inn's nightly cost before the tax is added.

Exercises 41–42 involve markup, the amount added to the dealer's cost of an item to arrive at the selling price of that item.

41. The selling price of a refrigerator is $584. If the markup is 25% of the dealer's cost, what is the dealer's cost of the refrigerator?

42. The selling price of a scientific calculator is $15. If the markup is 25% of the dealer's cost, what is the dealer's cost of the calculator?

In Exercises 43–60, solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe?

43.

44.

45. bh for b

46. Bh for B

47. for P

48. for r

49. for m

50. for h

51. for m

52. for M

53. for a

54. for b

55. for r

56. for t

57. for x

58. for y



A = LW for L

D = RT for R

A = 12

V = 13

I = Prt

C = 2πr

E = mc2

V = π hr2

y = mx + b

P = C + MC

A = h (a + b)12

A = h (a + b)12

S = P + Prt

S = P + Prt

Ax + By = C

Ax + By = C








Page 375

59. for n

60. for d

Writing in Mathematics61. In your own words, describe a step-by-step approach for solving algebraic word problems.

62. Write an original word problem that can be solved using a linear equation. Then solve the problem.

63. Explain what it means to solve a formula for a variable.

64. Did you have difficulties solving some of the problems that were assigned in this Exercise Set? Discuss what you did if this happened to you. Did your course ofaction enhance your ability to solve algebraic word problems?

Critical Thinking ExercisesMake Sense? In Exercises 65–68, determine whether each statement makes sense or does not make sense, and explain your reasoning.

65. By modeling attitudes of college freshmen from 1969 through 2010, I can make precise predictions about the attitudes of the freshman class of 2020.


= + (n − 1)dan a1

= + (n − 1)dan a1








6 Algebra: Equations and Inequalities > 6.4 Linear Inequalities in One Variable

66. I find the hardest part in solving a word problem is writing the equation that models the verbal conditions.

67. I solved a formula for one of its variables, so now I have a numerical value for that variable.

68. After a 35% reduction, a computer's price is $780, so I determined the original price, x, by solving

69. The price of a dress is reduced by 40%. When the dress still does not sell, it is reduced by 40% of the reduced price. If the price of the dress after both reductionsis $72, what was the original price?

70. In a film, the actor Charles Coburn plays an elderly “uncle” character criticized for marrying a woman when he is 3 times her age. He wittily replies, “Ah, but in 20years time I shall only be twice her age.” How old is the “uncle” and the woman?

71. Suppose that we agree to pay you for every problem in this chapter that you solve correctly and fine you for every problem done incorrectly. If at the end of26 problems we do not owe each other any money, how many problems did you solve correctly?

72. It was wartime when the Ricardos found out Mrs. Ricardo was pregnant. Ricky Ricardo was drafted and made out a will, deciding that $14,000 in a savingsaccount was to be divided between his wife and his child-to-be. Rather strangely, and certainly with gender bias, Ricky stipulated that if the child were a boy, he wouldget twice the amount of the mother's portion. If it were a girl, the mother would get twice the amount the girl was to receive. We'll never know what Ricky was thinkingof, for (as fate would have it) he did not return from the war. Mrs. Ricardo gave birth to twins—a boy and a girl. How was the money divided?

73. A thief steals a number of rare plants from a nursery. On the way out, the thief meets three security guards, one after another. To each security guard, the thief isforced to give one-half the plants that he still has, plus 2 more. Finally, the thief leaves the nursery with 1 lone palm. How many plants were originally stolen?

In Exercises 74–75, solve each proportion for x.

74.

75.

Group Exercise76. One of the best ways to learn how to solve a word problem in algebra is to design word problems of your own. Creating a word problem makes you very aware ofprecisely how much information is needed to solve the problem. You must also focus on the best way to present information to a reader and on how much informationto give. As you write your problem, you gain skills that will help you solve problems created by others.

The group should design five different word problems that can be solved using linear equations. All of the problems should be on different topics. For example, thegroup should not have more than one problem on price reduction. The group should turn in both the problems and their algebraic solutions.

6.4 Linear Inequalities in One Variable

What am I Supposed to Learn?After you have read this section, you should be able to:

1 Graph subsets of real numbers on a number line.

2 Solve linear inequalities.

3 Solve applied problems using linear inequalities.

RENT-A-HEAP, A CAR RENTAL company, charges $125 per week plus $0.20 per mile to rent one of their cars. Suppose you are limited by how much money you canspend for the week: You can spend at most $335. If we let x represent the number of miles you drive the heap in a week, we can write an inequality that models thegiven conditions: Chapter 6 Algebra: Equations and Inequalities



x − 0.35 = 780.

8¢ 5¢

=x+a

a

b+c

c

=ax−b

b

c−d

d







Page 376


The weekly

charge of $125

▼

125

plus

▼

+

the charge of

$0.20 per mile

for x miles

▼

0.20x

must be less

than

or equal to

▼

≤

$335.

▼

335.






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6.3 Applications of Linear Equations - Moreland MATH Algebra: Equations and Inequalities > 6.3 Applications of Linear Equations > Solving a Formula for One of Its Variables `