Applications of Systems of Linear Equations

Transcription

1 5.2 Applications of Systems of Linear Equations 5.2 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 approached with the same five-step strategy presented here to begin the discussion. Step by Step: Solving Applications Step 1 Step 2 Step 3 Step 4 Step 5 Read the problem carefully to determine the unknown quantities. Choose a variable to represent any unknown. Translate the problem to the language of algebra to form a system of equations. Solve the system of equations, and answer the question of the original problem. Verify your solution by returning to the original problem. Example 1 Solving a Mixture Problem A coffee merchant has two types of coffee beans, one selling for $9 per pound and the other for $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 of the 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 based on the total amount of the mixture, the other on the mixture s value. NOTE Because we use two variables, we must form two equations. x y 100 The mixture must weigh 100 lb. (1) 9x 15y 1350 (2) 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 9 and add to eliminate x. 9x 9y 900 9x 15y y 450 y 75 lb 319

2 320 CHAPTER 5 SYSTEMS OF LINEAR RELATIONS 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. CHECK YOURSELF 1 Peanuts, which sell for $4.80 per pound, and cashews, which sell for $12 per pound, are to be mixed to form a 60-lb mixture selling for $6 per pound. How much of each type of nut should be used? A related problem is illustrated in Example 2. Example 2 Solving a Mixture Problem A chemist has a 25% and a 50% acid solution. How much of each solution should be used to form 200 ml of a 35% acid solution? x ml 25% y ml 50% 200 ml 35% Drawing a sketch of a problem is often a valuable part of the problem-solving strategy. 25% solution 50% solution 35% solution Step 1 The unknowns in this case are the amounts of the 25% and 50% solutions to be used in forming the mixture. Step 2 Again we use two variables to represent the two unknowns. Let x be the amount of the 25% solution and y the amount of the 50% solution. Let s draw a picture before proceeding to form a system of equations. Step 3 Now, to form our two equations, we want to consider two relationships: the total amounts combined and the amounts of acid combined. NOTE Total amounts combined. NOTE Amounts of acid combined. From our sketch of the problem, we have x y 200 (3) 0.25x 0.50y 0.35(200) (4) Step 4 Now, clear equation (4) of decimals by multiplying equation (4) by 100. The solution then proceeds as before, with the result x 120 ml (25% solution) y 80 ml (50% solution)

3 APPLICATIONS OF SYSTEMS OF LINEAR EQUATIONS SECTION 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. CHECK YOURSELF 2 A pharmacist wants to prepare 300 ml of a 20% alcohol solution. How much of a 30% solution and a 15% solution should be used to form the desired mixture? Applications that involve a constant rate of travel, or speed, require the use of the distance formula d rt in which d distance traveled r rate, or speed t time Example 3 illustrates this approach. 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. What is the rate of the boat in still water? What is the rate of the current? water current Step 1 Step 2 We want to find the two rates. Let x be the rate of the boat in still water and y the rate of the current. NOTE Downstream the rate is then x y Upstream, the rate is x y Step 3 To form a system, think about the following. Downstream, the rate of the boat is increased 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 that strategy here. d r t Downstream 36 x y 2 Upstream 36 x y 3 Because d rt, from the table we can easily form two equations: 36 (x y)(2) (5) 36 (x y)(3) (6)

4 322 CHAPTER 5 SYSTEMS OF LINEAR RELATIONS Step 4 We clear equations (5) and (6) of parentheses and simplify, to write the equivalent 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 downstream cases. We leave that to you. CHECK YOURSELF 3 A plane flies 480 mi in an easterly direction, with the wind, in 4 h. Returning westerly along the same route, against the wind, the plane takes 6 h. What is the rate of the plane in still air? What is the rate of the wind? Direction of travel Direction of wind The use of systems of equations in problem solving has many applications in a business setting. Example 4 illustrates one such application. Example 4 Solving a Business-Based Application A manufacturer produces a standard model and a deluxe model of a 25-inch (in.) television set. The standard model requires 12 h of labor to produce, and the deluxe model requires 18 h. The company has 360 h of labor available per week. The plant s capacity is a total of 25 sets per week. If all the available time and capacity are to be used, how many of each type of set should be produced? Step 1 The unknowns in this case are the number of standard and deluxe models that can be produced.

5 APPLICATIONS OF SYSTEMS OF LINEAR EQUATIONS SECTION NOTE The choices for x and y could have been reversed. 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 of sets that can be produced and the total labor hours available. x y 25 12x 18y 360 Labor hours standard sets Total number of sets Total labor hours available Labor hours deluxe sets 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 standard sets and 10 deluxe sets per week. Step 5 We leave the check of this result to the reader. CHECK YOURSELF 4 A manufacturer produces standard cassette players and compact disc players. The cassette players require 2 h of electronic assembly and the CD players 3 h. The cassette players require 4 h of case assembly and the CD players 2 h. The company has 120 h of electronic assembly time available per week and 160 h of case assembly time. How many of each type of unit can be produced each week if all available assembly time is to be used? Let s look at one final application that leads to a system of two equations. Example 5 Solving a Business-Based Application NOTE You first saw this type of linear model in exercises in Section 4.3. Two car rental agencies have the following rate structures for a subcompact car. Urent charges $50 per day plus 15 per mile. Painz charges $45 per day plus 20 per mile. If you rent a car for 1 day, for what number of miles will the two companies have the same total charge? Letting c represent the total a company will charge and m the number of miles driven, we calculate the following. For Urent: c m (7) For Painz: c m (8) The system can be solved most easily by substitution. Substituting m for c in equation (7) gives m m 0.05m 5 m 100 mi

6 324 CHAPTER 5 SYSTEMS OF LINEAR RELATIONS The graph of the system is shown below. c (cost) Urent Painz (100, 65) m (miles) From the graph, how would you make a decision about which agency to use? CHECK YOURSELF 5 For a compact car, the same two companies charge $54 per day plus 20 per mile and $51 per day plus 22 per mile. For a 2-day rental, when will the charges be the same? CHECK YOURSELF ANSWERS lb of peanuts and 10 lb of cashews ml of the 30% and 200 ml of the 15% mi/h plane and 20 mi/h wind cassette players and 20 CD players 5. At 300 mi, $168 charge

7 Name 5.2 Exercises Section Date Each application in exercises 1 to 8 can be solved by the use of a system of linear equations. Match the application with the appropriate system below. (a) 12x 5y 116 (b) x y x 12y x 0.09y 600 (c) x y 200 (d) x y x 0.60y 90 y 3x 4 (e) 2(x y) 36 (f) x y 200 3(x y) x 4.50y 980 (g) L 2W 3 (h) x y 120 2L 2W x 5.40y Number problem. One number is 4 less than 3 times another. If the sum of the numbers is 36, what are the two numbers? 2. Recreation. Suppose a movie theater sold 200 adult and student tickets for a showing with a revenue of $980. If the adult tickets were $6.50 and the student tickets were $4.50, how many of each type of ticket were sold? ANSWERS Geometry. The length of a rectangle is 3 cm more than twice its width. If the perimeter 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 per pound with cashews selling at $5.40 per pound to form 120 lb of a mixed-nut blend that 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 savings account paying 6% interest, and the other a bond paying 9%. If the annual interest from 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 be used 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 still water? What was the rate of the river s current? 325

8 ANSWERS In exercises 9 to 30, solve by choosing a variable to represent each unknown quantity and writing a system of equations. 9. Mixture problem. Suppose 750 tickets were sold for a concert with a total revenue of $5300. If adult tickets were $8 and student tickets were $4.50, how many of each type of ticket were sold? 10. Mixture problem. Theater tickets sold for $7.50 on the main floor and $5 in the balcony. The total revenue was $3250, and there were 100 more main-floor tickets sold 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 the perimeter 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 the perimeter 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 of that purchase was $25. Find the cost of a single bag of mulch and a single bag of fertilizer. 14. Mixture problem. The cost of an order for 10 computer disks and 3 packages of paper was $ The next order was for 30 disks and 5 packages of paper, and its cost was $ 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 one selling for $4 per pound and the other for $6.50 per pound. She wishes to blend the beans to form a 150-lb mixture that will sell for $4.75 per pound. How many pounds of each grade of bean should be used in the mixture? 16. Mixture problem. A candy merchant sells jelly beans at $3.50 per pound and gumdrops at $4.70 per pound. To form a 200-lb mixture that will sell for $4.40 per pound, how many pounds of each type of candy should be used? 326

9 ANSWERS 17. Investment. Cheryl decided to divide $12,000 into two investments one a time deposit that pays 8% annual interest and the other a bond that pays 9%. If her annual interest was $1010, how much did she invest at each rate? 18. Investment. Miguel has $3000 more invested in a mutual fund paying 5% interest than in a savings account paying 3%. If he received $310 in interest for 1 year, how much did he have invested in the two accounts? 19. Science. A chemist mixes a 10% acid solution with a 50% acid solution to form 400 ml of a 40% solution. How much of each solution should be used in the mixture? 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 should be used? 21. Motion. A boat traveled 36 mi up a river in 3 h. Returning downstream, the boat took 2 h. What is the boat s rate in still water, and what is the rate of the river s current? Motion. A jet flew east a distance of 1800 mi with the jetstream in 3 h. Returning west, against the jetstream, the jet took 4 h. Find the jet s speed in still air and the rate of the jetstream. 23. Number problem. The sum of the digits of a two-digit number is 8. If the digits are reversed, the new number is 36 more than the original number. Find the original number. Hint: If u represents the units digit of the number and t the tens digit, the original number can be represented by 10t u. 24. Number problem. The sum of the digits of a two-digit number is 10. If the digits are reversed, the new number is 54 less than the original number. What was the original 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 solar model 15 min. There are 450 min of assembly time available per day. Both models require 8 min for packaging, and 280 min of packaging time are available per day. If the manufacturer wants to use all the available time, how many of each unit should be produced per day? 327

10 ANSWERS 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? 27. Economics. In economics, a demand equation gives the quantity D that will be demanded by consumers at a given price p, in dollars. Suppose that D 210 4p for a particular product. A supply equation gives the supply S that will be available from producers at price p. Suppose also that for the same product S 10p. The equilibrium point is that point at which the supply equals the demand (here, where S D). Use the given equations to find the equilibrium point. 28. Economics. Suppose the demand equation for a product is D 150 3p and the supply equation is S 12p. Find the equilibrium point for the product. 29. Consumer affairs. Two car rental agencies have the following rate structure for compact cars. Company A: $30/day and 22 /mi. Company B: $28/day and 26 /mi. For a 2-day rental, at what number of miles will the charges be the same? 30. Construction. Two construction companies submit the following bid. Company A: $5000 plus $15/square foot of building. Company B: $7000 plus $12.50/square foot of building. For what number of square feet of building will the bids of the two companies be the same? Certain systems that are not linear can be solved with the methods of this section if we first substitute to change variables. For instance, the system 1 x 1 y 4 1 x 3 y can be solved by the substitutions u and v. That gives the system u v 4 and x y u 3v 6. The system is then solved for u and v, and the corresponding values for x and y are found. Use this method to solve the systems in exercises 31 to

11 ANSWERS x 3 x 1 y 4 y 1 1 x 3 y x 3 y 4 2 x 6 10 y 4 x 3 y 3 4 x 3 y 1 12 x 1 y 1 Writing the equation of a line through two points can be done by the following method. Given the coordinates of two points, substitute each pair of values into the equation y mx b. This gives a system of two equations in variables m and b, which can be solved as before. In exercises 35 and 36, write the equation of the line through each of the following pairs of points, using the method outlined above (2, 1) and (4, 4) 36. ( 3, 7) and (6, 1) In exercises 37 and 38, use your calculator to approximate the solution to each system. Express your answer to the nearest tenth. 37. y 2x x 4y 7 2x 3y 1 2x 3y 1 y y x x 329

12 ANSWERS For exercises 39 and 40, adjust the viewing window on your calculator so that you can see the point of intersection for the two lines representing the equations in the system. Then approximate the solution x 12y x 3y 10 7x 2y 44 x 5y 58 y y x x Find values for m and b in the following system so that the solution to the system is (1, 2). mx 3y 8 3x 4y b 42. Find values for m and b in the following system so that the solution to the system is ( 3, 4). 5x 7y b mx y Complete the following statements in your own words: To solve an equation means to.... To solve a system of equations means to A system of equations such as the one below is sometimes called a 2-by-2 system of linear equations. 3x 4y 1 x 2y 6 Explain this term. 330

13 ANSWERS 45. Complete this statement in your own words: All the points on the graph of the equation 2x 3y 6... Exchange statements with other students. Do you agree with other students statements? Does a system of linear equations always have a solution? How can you tell without graphing that a system of two equations will be graphed as two parallel lines? Give some examples to explain your reasoning. 47. (a) (b) (c) 47. Suppose we have the following linear system: Ax By C (1) Dx Ey F (2) (a) Write the slope of the line determined by equation (1). (b) Write the slope of the line determined by equation (2). (c) What must be true about the given coefficients to guarantee that the system is consistent? (a) (b) 48. We have discussed three different methods of solving a system of two linear equations in two unknowns: the graphical method, the addition method, and the substitution method. Discuss the strengths and weaknesses of each method. 49. Determine a system of two linear equations for which the solution is (3, 4). Are there other systems that have the same solution? If so, determine at least one more and explain why this can be true. 50. Suppose we have the following linear system: Ax By C (1) Dx Ey F (2) (a) Multiply equation (1) by D, multiply equation (2) by A and add. This will allow you to eliminate x. Solve for y and indicate what must be true about the coefficients for a unique value for y to exist. (b) Now return to the original system and eliminate y instead of x. (Hint: try multiplying equation (1) by E and equation (2) by B.) Solve for x and again indicate what must be true about the coefficients for a unique value for x to exist. 331

14 Answers 1. (d) 3. (g) 5. (h) 7. (c) adult, 200 student tickets in. 15 in. 13. Mulch: $1.80; fertilizer: $ lb of $4 beans, 45 lb of $6.50 beans 17. $7000 time deposit, $5000 bond ml of 10%, 300 ml of 50% mi/h boat, 3 mi/h current battery powered, 20 solar models 27. p mi 2 3, (1.3, 0.5) 39. (6, 2) y 1 3, 3 2 y 2x 3 y 3 2 x 2 y 25 y x 3 x x 2 y 3 x 41. m 2, b y 2 x 22 A D 47. (a) ; (b) ; (c) AE BD B E 332