Applications of Systems of Linear Equations

OBJECTIVE

1. Use a system of equations to solve an application

We are now ready to apply our equation-solving skills to solving various applications or word problems. Being able to extend these skills to problem solving is an important goal, and the procedures developed here are used throughout the rest of the book.

Although we consider applications from a variety of areas in this section, all are ap-proached with the same five-step strategy presented here to begin the discussion.

Step 1 Read the problem carefully to determine the unknown quantities.Step 2 Choose a variable to represent any unknown.Step 3 Translate the problem to the language of algebra to form a system of

equations.Step 4 Solve the system of equations, and answer the question of the original

problem.Step 5 Verify your solution by returning to the original problem.

Step by Step: Solving Applications

Solving a Mixture Problem

A coffee merchant has two types of coffee beans, one selling for $9 per pound and the otherfor $15 per pound. The beans are to be mixed to provide 100 lb of a mixture selling for$13.50 per pound. How much of each type of coffee bean should be used to form 100 lb ofthe mixture?

Step 1 The unknowns are the amounts of the two types of beans.

Step 2 We use two variables to represent the two unknowns. Let x be the amount of$9 beans and y the amount of $15 beans.

Step 3 We now want to establish a system of two equations. One equation will be basedon the total amount of the mixture, the other on the mixture’s value.

x � y � 100 The mixture must weigh 100 lb.

9x � 15y � 100 (13.50)

Value of Value of Total value$9 beans $15 beans

Step 4 An easy approach to the solution of the system is to multiply equation (1) by �9and add to eliminate x.

�9x � 9y � �900

9x � 15y � 1350

6y � 450

y � 75 lb

Example 1

NOTE Because we use twovariables, we must form twoequations.

By substitution in equation (1), we have

x � 25 lb

Step 5 To check the result, show that the value of the $9 beans, added to the value of the$15 beans, equals the desired value of the mixture.

Solving

A chemist has a 25% and a 50% acid solution. How much of each solution should be usedto form 200 mL of a 35% acid solution?

Step 1 The unknowns in this case are the amounts of the 25% and 50% solutions to beused in forming the mixture.

Step 2 Again we use two variables to represent the two unknowns. Let x be the amountof the 25% solution and y the amount of the 50% solution. Let’s draw a picture before pro-ceeding to form a system of equations.

Step 3 Now, to form our two equations, we want to consider two relationships: the totalamounts combined and the amounts of acid combined.

From our sketch of the problem, we have

x � y � 200

0.25x � 0.50y � 0.35(200)

Step 4 Now, clear equation (2) of decimals by multiplying equation (2) by 100. The solution then proceeds as before, with the result

x � 120 mL (25% solution)

y � 80 mL (50% solution)

25% solution

35% solution

50% solution

NOTE Total amountscombined.

NOTE Amounts of acidcombined.

200 mL35%

y mL50%

x mL25%

� �

Drawing a sketch of a problem isoften a valuable part of theproblem-solving strategy.

Exampe 2

Step 5 To check, show that the amount of acid in the 25% solution, (0.25)(120), added to the amount in the 50% solution, (0.50)(80), equals the correct amount in the mixture,(0.35)(200). We leave that to you.

Applications that involve a constant rate of travel, or speed, require the use of the dis-tance formula

d � rt

in which d � distance traveledr � rate, or speedt � time

Example 3

Solving a Distance-Rate-Time Problem

A boat can travel 36 mi downstream in 2 h. Coming back upstream, the boat takes 3 h. Whatis the rate of the boat in still water? What is the rate of the current?

Step 1 We want to find the two rates.

Step 2 Let x be the rate of the boat in still water and y the rate of the current.

Step 3 To form a system, think about the following. Downstream, the rate of the boat isincreased by the effect of the current. Upstream, the rate is decreased.

In many applications, it helps to lay out the information in tabular form. Let’s try thatstrategy here.

Because d � rt, from the table we can easily form two equations:

36 � (x � y)(2)

36 � (x � y)(3)

d r t

Downstream 36 x � y 2Upstream 36 x � y 3

water current

NOTE Downstream the rate is x � yUpstream, the rate is x � y

Step 4 We clear equations (1) and (2) of parentheses and simplify, to write the equiva-lent system

x � y � 18

x � y � 12

Solving, we have

x � 15 mi/h

y � 3 mi/h

Step 5 To check, verify the d � rt equation in both the upstream and the downstreamcases. We leave that to you.

Example 4

Solving a Business-Based Application

A manufacturer produces a standard model and a deluxe model of a 25-inch (in.) televisionset. The standard model requires 12 h of labor to produce, and the deluxe model requires18 h. The company has 360 h of labor available per week. The plant’s capacity is a total of25 sets per week. If all the available time and capacity are to be used, how many of eachtype of set should be produced?

Step 1 The unknowns in this case are the number of standard and deluxe models that canbe produced.

Step 2 Let x be the number of standard models and y the number of deluxe models.

Step 3 Our system will come from the two given conditions that fix the total number ofsets that can be produced and the total labor hours available.

x � y � 25

12x � 18y � 360

Step 4 Solving the system in step 3, we have

x � 15 and y � 10

which tells us that to use all the available capacity, the plant should produce 15 standardsets and 10 deluxe sets per week.

Step 5 Check the results and State the conclusion.

Total numberof sets

Total laborhours available

Labor hours—deluxe sets

NOTE The choices for x and ycould have been reversed.

Labor hours—standard sets

ExercisesEach application in exercises 1 to 8 can be solved by the use of a system of linearequations. Match the application with the appropriate system below.

(a) 12x � 5y � 116 (b) x � y � 80008x � 12y � 112 0.06x � 0.09y � 600

(c) x � y � 200 (d) x � y � 360.20x � 0.60y � 90 y � 3x � 4

(e) 2(x � y) � 36 (f) x � y � 2003(x � y) � 36 6.50x � 4.50y � 980

(g) L � 2W � 3 (h) x � y � 1202L � 2W � 36 2.20x � 5.40y � 360

1. Number problem. One number is 4 less than 3 times another. If the sum of thenumbers is 36, what are the two numbers?

2. Recreation. Suppose a movie theater sold 200 adult and student tickets for ashowing with a revenue of $980. If the adult tickets were $6.50 and the studenttickets were $4.50, how many of each type of ticket were sold?

3. Geometry. The length of a rectangle is 3 cm more than twice its width. If theperimeter of the rectangle is 36 cm, find the dimensions of the rectangle.

4. Business. An order of 12 dozen roller-ball pens and 5 dozen ballpoint pens cost$116. A later order for 8 dozen roller-ball pens and 12 dozen ballpoint pens cost$112. What was the cost of 1 dozen of each type of pen?

5. Mixture problem. A candy merchant wants to mix peanuts selling at $2.20 perpound with cashews selling at $5.40 per pound to form 120 lb of a mixed-nut blendthat will sell for $3 per pound. What amount of each type of nut should be used?

6. Investment. Donald has investments totaling $8000 in two accounts—one a savingsaccount paying 6% interest, and the other a bond paying 9%. If the annual interestfrom the two investments was $600, how much did he have invested at each rate?

7. Mixture problem. A chemist wants to combine a 20% alcohol solution with a 60%solution to form 200 mL of a 45% solution. How much of each solution should beused to form the mixture?

8. Motion problem. Xian was able to make a downstream trip of 36 mi in 2 h.Returning upstream, he took 3 h to make the trip. How fast can his boat travel in stillwater? What was the rate of the river’s current?

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

In exercises 9 to 30, solve by choosing a variable to represent each unknown quantity andwriting a system of equations.

9. Mixture problem. Suppose 750 tickets were sold for a concert with a total revenueof $5300. If adult tickets were $8 and student tickets were $4.50, how many of eachtype of ticket were sold?

10. Mixture problem. Theater tickets sold for $7.50 on the main floor and $5 in thebalcony. The total revenue was $3250, and there were 100 more main-floor ticketssold than balcony tickets. Find the number of each type of ticket sold.

11. Geometry. The length of a rectangle is 3 in. less than twice its width. If theperimeter of the rectangle is 84 in., find the dimensions of the rectangle.

12. Geometry. The length of a rectangle is 5 cm more than 3 times its width. If theperimeter of the rectangle is 74 cm, find the dimensions of the rectangle.

13. Mixture problem. A garden store sold 8 bags of mulch and 3 bags of fertilizer for$24. The next purchase was for 5 bags of mulch and 5 bags of fertilizer. The cost ofthat purchase was $25. Find the cost of a single bag of mulch and a single bag offertilizer.

14. Mixture problem. The cost of an order for 10 computer disks and 3 packages ofpaper was $22.50. The next order was for 30 disks and 5 packages of paper, and itscost was $53.50. Find the price of a single disk and a single package of paper.

15. Mixture problem. A coffee retailer has two grades of decaffeinated beans—oneselling for $4 per pound and the other for $6.50 per pound. She wishes to blend thebeans to form a 150-lb mixture that will sell for $4.75 per pound. How many poundsof each grade of bean should be used in the mixture?

16. Mixture problem. A candy merchant sells jelly beans at $3.50 per pound andgumdrops at $4.70 per pound. To form a 200-lb mixture that will sell for $4.40 perpound, how many pounds of each type of candy should be used?

ANSWERS

9.

10.

11.

12.

13.

14.

15.

16.

17. Investment. Cheryl decided to divide $12,000 into two investments—one a timedeposit that pays 8% annual interest and the other a bond that pays 9%. If her annualinterest was $1010, how much did she invest at each rate?

18. Investment. Miguel has $3000 more invested in a mutual fund paying 5% interestthan in a savings account paying 3%. If he received $310 in interest for 1 year, howmuch did he have invested in the two accounts?

19. Science. A chemist mixes a 10% acid solution with a 50% acid solution to form400 mL of a 40% solution. How much of each solution should be used in themixture?

20. Science. A laboratory technician wishes to mix a 70% saline solution and a 20%solution to prepare 500 mL of a 40% solution. What amount of each solution shouldbe used?

21. Motion. A boat traveled 36 mi up a river in 3 h. Returning downstream, the boattook 2 h. What is the boat’s rate in still water, and what is the rate of the river’scurrent?

22. Motion. A jet flew east a distance of 1800 mi with the jetstream in 3 h. Returningwest, against the jetstream, the jet took 4 h. Find the jet’s speed in still air and the rateof the jetstream.

23. Number problem. The sum of the digits of a two-digit number is 8. If the digits arereversed, the new number is 36 more than the original number. Find the originalnumber. Hint: If u represents the units digit of the number and t the tens digit, theoriginal number can be represented by 10t � u.

24. Number problem. The sum of the digits of a two-digit number is 10. If the digitsare reversed, the new number is 54 less than the original number. What was theoriginal number?

25. Business. A manufacturer produces a battery-powered calculator and a solar model.The battery-powered model requires 10 min of electronic assembly and the solarmodel 15 min. There are 450 min of assembly time available per day. Both modelsrequire 8 min for packaging, and 280 min of packaging time are available per day. Ifthe manufacturer wants to use all the available time, how many of each unit should beproduced per day?

ANSWERS

17.

18.

19.

20.

21.

22.

23.

24.

25.

26. Business. A small tool manufacturer produces a standard- and a cordless-model power drill. The standard model takes 2 h of labor to assemble and the cordless model 3 h. There are 72 h of labor available per week for the drills. Material costs for the standard drill are $10, and for the cordless drill they are $20. The company wishes to limit material costs to $420 per week. How many of each model drill should be produced to use all the available resources?

26.

Answers1. (d) 3. (g) 5. (h) 7. (c) 9. 550 adult, 200 student tickets11. 27 in. � 15 in. 13. Mulch: $1.80; fertilizer: $3.2015. 105 lb of $4 beans, 45 lb of $6.50 beans 17. $7000 time deposit, $5000 bond19. 100 mL of 10%, 300 mL of 50% 21. 15 mi/h boat, 3 mi/h current23. 26 25. 15 battery powered, 20 solar models