More with Systems of Equations

Transcription

1 More with Systems of Equations In 2008, 4.7 million Americans went on a rafting expedition. In Georgia, outfitters run whitewater expeditions for ages 8 and up on the Chattooga River Systems of Equations Using Linear Combinations to Solve a Linear System What s for Lunch? Solving More Systems Making Decisions Using the Best Method to Solve a Linear System Going Green Using a Graphing Calculator to Solve Linear Systems Beyond the Point of Intersection Using a Graphing Calculator to Analyze a System

2 646 Chapter 12 More with Systems of Equations

3 Systems of Equations Using Linear Combinations to Solve a Linear System Learning Goals In this lesson, you will: Write a system of equations to represent a problem context. Solve a system of equations algebraically using linear combinations (elimination). Key Term linear combinations method (elimination) Morse code is a communication system which allows people to speak with sound. Words are transmitted using short sounds called dits, which are represented in writing as dots, and long sounds, called dahs, which are represented in writing as dashes. The letters of the alphabet and digits each have their own unique collection of dits and dahs: A B C D E F G H I J K L M N O P Q R S T When you combine these codes, you can produce sentences in Morse code. U V W X Y Z Try it out! Communicate with your friends using Morse code Using Linear Combinations to Solve a Linear System 647

4 Problem 1 End of Year Trip There are a total of 324 students enrolled in Armstrong Elementary School. The girls outnumber the boys by Write an equation in standard form that represents the total number of students at Armstrong Elementary School. Use x to represent the number of girls, and use y to represent the number of boys. 2. Write an equation in standard form to represent the number of girls in relationship to the number of boys. 3. How are these equations the same? 4. How are the equations different? 5. Complete parts (a) through (f) to write and solve a linear system of equations for this situation. a. Write a linear system for this problem situation. b. Add the two equations together. c. Solve the resulting equation. d. Substitute the value of x that you obtained in part (c) into one of the original equations and solve to determine the value of y. I see how you can add equations. (4 + 2) = 6 (4-2) = 2 So, if I can add 6 and 2 and get 8, then that means I can add (4 + 2) and (4-2) and get 8 also. So, (4 + 2) + (4-2) = Chapter 12 More with Systems of Equations

5 e. What is the solution of your linear system? f. Check your solution algebraically. 6. Check your solution by creating a graph of your linear system on the coordinate plane shown. Choose your variables, bounds, and intervals. Be sure to label your graph. Variable Quantity Lower Bound Upper Bound Interval y Number of Boys at Armstrong Elementary x Number of Girls at Armstrong Elementary 7. Interpret the solution of the linear system in this problem situation. 8. What effect did adding the equations together have? 9. Describe how the coefficients of y in the original system are related Using Linear Combinations to Solve a Linear System 649

6 Problem 2 Hotel Promotion The Splash and Stay Resort has an indoor water park, an arcade, a lounge, and restaurants. The resort is offering two winter specials. The first special is two nights and four meals for $270, while the second is three nights and eight meals for $ Write an equation in standard form that represents the first resort special. Let n represent the cost for one night at the resort, and let m represent the cost for each meal. 2. Write an equation in standard form that represents the second resort special. 3. How are these equations the same? 4. How are these equations different? 5. Multiply each side of the equation that represents the first resort special by 22. Simplify the equation; maintain standard form. If I multiply both sides of an equation by the same number, is the equation still true? 650 Chapter 12 More with Systems of Equations

7 6. Write a linear system of equations using the transformed equation that represents the first special and the equation that represents the second special. a. How do the coefficients of the equations in your linear system of equations compare? b. Add the equations in your linear system together. Then, simplify the result. What does the result represent? c. How will you determine the m value of the linear system? When you divide a negative value by -1, you make it positive. d. Determine the value of m for the linear system. e. What is the solution of the linear system? f. Interpret the solution of the linear system in the problem situation. g. Check your solution algebraically Using Linear Combinations to Solve a Linear System 651

8 Problem 3 Linear Combinations The method you used to solve the linear systems in Problems 1 and 2 is called the linear combinations method. The linear combinations method is a process to solve a system of equations by adding two equations to each other, resulting in an equation with one variable. You can then determine the value of one variable and use it to find the value of the other variable. In many cases, one, or both, of the equations in the system must be multiplied by a constant so that when the equations are added together, the result is an equation in one variable. This means that the coefficients of either the term containing x or y must be opposites. For example, consider this system: 4x 1 2y 5 3 5x 1 3y 5 4 You can multiply the first equation by 3 and the second equation by 22 to produce opposite coefficients for y that will eliminate each other. Alternatively, you can multiply the first equation by 23 and the second equation by Multiply the first equation by 3 and multiply the second equation by 22. Then, rewrite each equation. 2. Solve the new linear system. Show your work. Did you check your solution by substituting the ordered pair back into the original equations? 652 Chapter 12 More with Systems of Equations

9 3. For each linear system shown, describe the first step you would take to solve the system using the linear combination method. Identify the variable that will be solved for when you add the equations. a. 5x 1 2y 5 10 and 3x 1 2y 5 6 b. x 1 3y 5 15 and 5x 1 2y 5 7 c. 4x 1 3y 5 12 and 3x 1 2y Using Linear Combinations to Solve a Linear System 653

10 4. Solve each system using linear combinations. a. 2x 1 y 5 8 3x 2 y 5 7 4x 1 3y 5 24 b. 3x 1 y x 1 5y 5 17 c. 2x 1 3y 5 11 Be prepared to share your solutions and methods. 654 Chapter 12 More with Systems of Equations

11 What s for Lunch? Solving More Systems Learning Goals In this lesson, you will: Write a linear system of equations to represent a problem context. Choose the best method to solve a linear system of equations. Suppose one cell phone company charges $0.10 per minute for phone calls. Another company charges $60 per month for unlimited call time. Can you write a system of equations to compare the two plans? Which one would be best for you? Which one would be best for your parents? 12.2 Solving More Systems 655

12 Problem 1 Fundraising One day each month the Family and Consumer Science classes offer a deli lunch for the faculty and staff of the school to purchase. The staff has a choice of either a chef salad for $5.75 or a hoagie for $5. Today the Family and Consumer Science classes sold 85 lunches for a total of $464. Determine how many chef salads and hoagies were sold. 1. Write an equation in standard form that gives the total number of lunches in terms of the number of chef salads sold and the number of hoagies sold. Let x represent the number of chef salads sold, and let y represent the number of hoagies sold. 2. Write an equation in standard form that represents the amount of money collected. Use the same variables as those used in Question Write a system of linear equations to represent this problem situation. 4. What methods can you use to solve this system of linear equations? Think about all the strategies you have used in previous lessons. 656 Chapter 12 More with Systems of Equations

13 5. Determine the solution of this linear system of equations by using linear combinations and check your answer. 6. Interpret your solution to the linear system in terms of the problem situation. Problem 2 More Fundraising The Jewelry Club is making friendship bracelets with the school colors to sell in the school store. The bracelets are black and orange and come in two lengths: 5 inches and 7 inches. The club has enough beads to make a total of 84 bracelets. They have made 49 bracelets, which represents 1 the number of 5-inch bracelets plus 3 the number of 7-inch bracelets 2 4 they plan to make and sell. 1. Write an equation in standard form to represent the total number of bracelets the Jewelry Club can make out of the beads that they have. Let x represent the number of 5-inch bracelets, and let y represent the number of 7-inch bracelets. 2. Write an equation in standard form to represent the number of bracelets the Jewelry Club has made so far. Use the same variables as those used in Question Solving More Systems 657

14 3. Write a system of linear equations that represents the problem situation. 4. Karyn says that the first step she would take to solve this system would be to first multiply the second equation by the least common denominator (LCD) of the fractions. Is she correct? Explain your reasoning. 5. Rewrite the equation containing fractions as an equivalent equation without fractions. 6. Determine the solution to the system of equations by using linear combinations and check your answer. 7. Interpret the solution of the linear system in terms of the problem situation. 658 Chapter 12 More with Systems of Equations

15 8. Solve each linear system using linear combinations. Check all solutions. a. x 1 2y 5 2 5x 2 3y b. 1 x y 5 3 3x 1 5y Solving More Systems 659

16 0.6x 1 0.2y c. 0.5x 2 0.2y y x y 5 17 d. 1 2 x Chapter 12 More with Systems of Equations

17 Talk the Talk You have used three different methods for solving systems of equations: graphing, substitution, and linear combinations. 1. Describe how to use each method and the characteristics of the system that makes this method most appropriate. a. Graphing Method: b. Substitution Method: c. Linear Combinations Method: Which method do you like best? Be prepared to share your solutions and methods Solving More Systems 661

18 662 Chapter 12 More with Systems of Equations

19 Making Decisions Using the Best Method to Solve a Linear System Learning Goals In this lesson, you will: Write a linear system of equations to represent a problem context. Choose the best method to solve a linear system of equations. Problem 1 Roller Skating, Here We Come! The activities director of the Community Center is planning a skating event for all the students at the local middle school. There are several skating rinks in the area, but the director does not know which one to use. Skate Fest charges a fee of $200 plus $3 per skater, while Roller Rama charges $5 per skater. 1. Write an equation that gives the total cost of renting Skate Fest for the skating event in terms of the number of students attending. Define your variables. 2. Write an equation that gives the total cost of renting Roller Rama for the skating event in terms of the number of students attending. Use the same variables you used in Question Suppose the activities director anticipates that 50 students will attend. a. Calculate the total cost of using Skate Fest. b. Calculate the total cost of using Roller Rama Using the Best Method to Solve a Linear System 663

20 4. Suppose the activities director has $650 to spend on the skating event. a. Determine the number of students who can attend if the event is held at Skate Fest. b. Determine the number of students who can attend if the event is held at Roller Rama. 5. Write a system of equations to represent this problem situation. 6. When is the cost of each skating rink the same? Use an algebraic method to explain your reasoning. 664 Chapter 12 More with Systems of Equations

21 7. Which algebraic method did you use in Question 6? Explain your reasoning. 8. Complete the table of values to show the cost for different numbers of students attending the event at each rink. Quantity Name Number of Students Skate Fest Roller Rama Unit Expression x Is your solution confirmed by the table? Explain your reasoning Using the Best Method to Solve a Linear System 665

22 10. Check your solution by creating a graph of your system of equations. First, choose your bounds and intervals. Be sure to label the graph. Variable Quantity Lower Bound Upper Bound Interval A graph is sometimes not as exact as I'd like it to be. 11. Is your solution confirmed by your graph? 12. Which skating rink would you recommend to the activities director? Explain your reasoning. 666 Chapter 12 More with Systems of Equations

23 Problem 2 Another Consideration Super Skates offers the use of the rink for a flat fee of $1000 for an unlimited number of skaters. 1. Write a linear equation to represent this situation. Then, add the graph of this equation to the grid in Problem 1, Question Describe when going to Super Skates is a better option than going to Skate Fest or Roller Rama. 3. The activities director s final budget is $895, and she has chosen to host the event at Skate Fest. a. Write an inequality that represents this situation. b. Solve the inequality. c. What does this solution mean in terms of the problem situation? Be prepared to share your solutions and methods Using the Best Method to Solve a Linear System 667

24 668 Chapter 12 More with Systems of Equations

25 Going Green Using a Graphing Calculator to Solve Linear Systems Learning Goal In this lesson, you will: Write a linear system of equations to represent a problem context. The world's first graphing calculator was introduced in October of This graphing calculator contained 422 bytes of memory and could calculate to a precision of 13 digits. Some of the graphing calculators available in 2011 can show even three-dimensional graphs and have 2.5 megabytes of data over 2.5 million bytes! That's over 6000 times more memory than the first graphing calculator! 12.4 Using a Graphing Calculator to Solve Linear Systems 669

26 Problem 1 When HEVs Take Over the Market In 2008, about 16,500 hybrid cars were sold. In 2009, about 20,000 hybrids were sold. 1. What is the rate of change in the number of hybrids sold per year from 2008 to 2009? 2. Write an equation that gives the total number of hybrid cars sold since Assume that the rate of change in sales is constant. Define the variables. 3. In 2008, 13.2 million conventional autos were sold in the United States. In 2009, 11.4 million conventional autos were sold. What is the rate of change in the number of conventional automobiles sold in the United States per year from 2008 to 2009? 4. Write an equation that gives the total number of conventional cars sold since Assume that the rate of change in sales is constant. Define the variables. 670 Chapter 12 More with Systems of Equations

27 5. Write a system of linear equations that represents the total sale of hybrid cars and the total sale of conventional automobiles since If these trends were to continue, could the hybrid sales ever equal the conventional auto sales? Use what you know about the equations of a linear system to explain your answer. Problem 2 Using a Graphing Calculator There are many ways to use a graphing calculator to solve systems. You will complete a table of values and then graph the system of equations. 1. Follow the steps shown to use the table features of your graphing calculator. Step 1: Press Y5 and enter the system of equations. y , x y ,200, ,800,000x 12.4 Using a Graphing Calculator to Solve Linear Systems 671

28 Step 2: Set up the table. Press TBLSET (press 2nd and press WINDOW ). TblStart is the start of your table. The range of your independent values will be 1 to 20, so enter TblStart 5 0. Tbl is the increment (read delta table ). This value tells the table how to count. Enter Tbl 5 1. Then, the independent values in your table will appear as 1, 2, 3, and so on. Step 3: Press TABLE (press 2nd GRAPH ). Be sure the "Auto" option is selected for both "Indpnt" and "Depend" in the TABLE SETUP. Step 4: Use the down arrow keys to scroll through the table. Notice that the dependent values are in scientific notation. a. Complete the table of values that shows the sales of hybrid cars and conventional cars for different numbers of years. Quantity Name Years Since 2008 Hybrid Sales Conventional Cars Unit Year Cars Cars Expression x Chapter 12 More with Systems of Equations

29 b. Determine whether the sales of hybrids will ever equal the sales of conventional cars. Explain your reasoning by using the table of values. 2. Follow the steps shown to graph the system of equations using your graphing calculator and to determine the point of intersection. Step 1: Step 2: Step 3: Step 4: Press Y5 and enter y , x y ,200, ,800,000x Press WINDOW to set the bounds and intervals for the graph. You can use the table of values you completed in Question 1 to determine how to set the window. Xmin 5 0, Xmax 5 20, Xscl 5 5, Ymin 5 0, Ymax 5 600,000, and Yscl 5 100,000. Press GRAPH. The intersection of both lines should be visible. If it is not, go back and adjust the WINDOW settings. Determine the point of intersection; you must be able to view the intersection point. Press CALC (Press 2ND TRACE ). Press 5 or select 5 : intersect. Move the cursor toward the intersection point and press ENTER three times. The point of intersection will be displayed at the bottom of your screen. a. What is the point of intersection? b. What does this point of intersection mean in terms of the problem situation? 12.4 Using a Graphing Calculator to Solve Linear Systems 673

30 c. In what year would the sale of hybrids equal the sale of conventional cars? d. What can you predict about the sales of conventional cars from your graph? Talk the Talk Solve each system using a graphing calculator. y x 1 9 y 5 x x 1 3y x 1 2y 5 7 What predictions can you make about each solution by analyzing the equations first? 6x 2 8y x 2 4y 5 8 y x y x Be prepared to share your solutions and methods. 674 Chapter 12 More with Systems of Equations

31 Beyond the Point of Intersection Using a Graphing Calculator to Analyze a System Learning Goals In this lesson, you will: Write a linear system of equations to represent a problem context. Use a graphing calculator to solve a linear system of equations. Problem 1 What s the Count? Cinemaplex, the local movie theater, had a blockbuster weekend! On Friday, 218 people attended the matinee and 753 people attended the evening shows, bringing in a total of $8620. Saturday was even more profitable. There were 847 people who attended the matinee while 1215 people attended the evening movies, bringing in sales of $16, Write an equation in standard form that represents each night. Define x as the price of matinee movies, and define y as the price of evening movies. 2. Without solving, interpret the solution to the linear system of equations. 3. Determine the solution of the linear system using your graphing calculator. a. Rewrite each equation by solving for y. Do not perform the division. Friday: y 1 5 Saturday: y 2 5 b. Enter the equations into your graphing calculator Using a Graphing Calculator to Analyze a System 675

32 c. What is the point of solution? Explain how you know. d. Interpret the solution of this linear system of equations. Problem 2 Job Offers 1. Alex is applying for positions at two different electronic stores in neighboring towns. The first job offer is a $200 weekly salary plus 5% commission on sales. The second job offer is a $75 weekly salary plus 10% commission. a. Write a system of equations that represents the problem situation. Describe the variables. b. Without solving the system of linear equations, interpret the solution. c. Solve the system of equations using your graphing calculator. d. Interpret the solution of the system in terms of the problem situation. 676 Chapter 12 More with Systems of Equations

33 It is important to understand the point of intersection in this situation. Let s use the graphing calculator to analyze beyond the point of intersection. 2. Determine values for each graph by using the TRACE function on your graphing calculator. You may have to adjust the WINDOW. a. Determine how much money Alex would earn at the first store if he sold $3000. First, set Xmax and Ymax Press GRAPH. Then, press TRACE. Move the cursor to y 1. Enter 3000 and press ENTER. b. Determine how much money Alex would earn at the second store if he sold $3000. Move the cursor to y 2. Enter 3000 and press ENTER. c. What is the difference in the weekly pay between stores if Alex sells $3000? 3. What is the difference in pay if he sold $4225 weekly? Alex s sales targets for each job would be between $1500 and $3000 weekly. Each manager told Alex the same thing: Some weeks are better than others, depending on the time of year and the new releases of technology. 4. Which job offer would you recommend Alex take? Explain your reasoning Using a Graphing Calculator to Analyze a System 677

34 Talk the Talk Solve each linear system using the substitution method, linear combinations, or a graphing calculator. y 5 5x y 5 9x 2 4 Look at the structure of each system before you choose your solution method. 15x 1 3y x 2 3y y x 3. 3x 1 2y x 1 28y x 1 23y Be prepared to share your solutions and methods. 678 Chapter 12 More with Systems of Equations

35 Chapter 12 Summary Key Term linear combinations method (elimination) (12.1) Solving a System of Equations Using the Linear Combinations Method The linear combinations method is a process used to solve a system of equations by adding two equations to each other so that they result in an equation with one variable. Then, you can determine the value of one variable and use it to determine the value of the other variable. In many cases, you may need to multiply one or both of the equations in the system by a constant so that when the equations are added together, the result is an equation in one variable. This means that the coefficients of the term containing either x or y must be opposites. Example 5x 1 2y 5 16 and 2x 1 6y (5x 1 2y) 5 23(16) 215x 2 6y x 1 2y x 1 6y x 1 6y (2) 1 2y x y x y 5 6 Check: 5(2) 1 2(3) 16 2(2) 1 6(3) The solution is (2, 3). x 5 2 y 5 3 Nice work! Keep it up! Chapter 12 Summary 679

36 Example 1 x y ( 1 x y 5 7 ) 1 x y ( 1 x y 5 1 ) 6x 1 4y x 2 4y x 1 4y x 1 4y (8) 1 4y x 2 4y y x y x y x 5 8 y 5 9 Check: 1 (8) (9) 7 1 (8) (9) The solution is (8, 9). Writing a Linear System of Equations to Represent a Problem Context When two or more linear equations define a relationship between quantities, they form a system of linear equations. Use the data in the problem to write two related equations. Then, evaluate the equations by substituting the given value for x or y to determine the value of the other variable. Interpret the solution in terms of the problem context. Example Ling needs to print flyers. Printer A will charge $30 plus $0.50 per flyer. Printer B will charge $15 plus $1 per flyer. Let x represent the cost per flyer and let y represent the total cost of the order. Printer A: y x Printer B: y x To determine how much each printer will charge to print 300 flyers, you can use substitution and evaluate each equation. Printer A: (300) Printer B: Printer A will charge $180 for 300 flyers, and Printer B will charge $315 for 300 flyers. 680 Chapter 12 More with Systems of Equations

37 Choosing the Best Method to Solve a Linear System of Equations Use graphing or substitution to determine the solution of a linear system of equations. Either graph the equations to determine their point of intersection, or use substitution to set the equations equal to one another. These methods work well when y is the same for both equations and the equations are in slope-intercept form. Interpret the solution in terms of the problem context. Example y 5 2x 2 5 and y 5 x 1 1 Substitution: Graphing: 2x x 1 1 x 5 6 y y 5 7 The solution is (6, 7). y x Writing and Solving an Inequality to Represent a Problem Context When a problem situation defines an upper or lower limit for the value of y, such as a budget, write the linear equation as an inequality rather than an equality. Solve as usual and interpret the solution in terms of the problem context. Example Kyle belongs to a DVD club. He pays $14 per month plus $2 per DVD. If he sets a budget for himself of $30 per month, you can determine how many DVDs he can buy each month x 30 2 x 16 x 8 Kyle can buy up to eight DVDs each month. Chapter 12 Summary 681

38 Writing a Linear System of Equations to Represent a Problem Context When two or more linear equations define a relationship between quantities, they form a system of linear equations. Use the data in the problem to write two related equations. Calculate the slope of each equation by determining the rate of change for each situation. Example Two cars are depreciating at different rates. Car A s value went from $28,000 when it was sold new in 2007 to $20,000 in Car B s value went from $34,000 when it was sold new in 2007 to $30,000 in , ,000 Car A: ; y 5 28, x 2 30, ,000 Car B: ; y 5 34, x 2 Choosing the Best Method to Solve a Linear System of Equations When the coefficients in a system of equations are not convenient for using substitution or linear combinations, you may need to use a graphing calculator to solve. Follow the steps to graph the system of equations using your graphing calculator, and then determine the point of intersection. Example 13x 1 5y x 2 19y 5 82 Step 1: Step 2: Step 3: Press Y5 and enter y x y x Press WINDOW to set the bounds and intervals for the graph. Press GRAPH. The intersection of both lines should be visible. If it is not, go back and adjust the WINDOW settings. Step 4: Determine the point of intersection; you must be able to view the intersection point. Press CALC. Press 5 or select 5: intersect. Move the cursor toward the intersection point and press ENTER three times. The point of intersection will be displayed at the bottom of your screen. The solution is (2.7, 21.9). 682 Chapter 12 More with Systems of Equations

39 Writing a Linear System of Equations to Represent a Problem Context When two or more linear equations define a relationship between quantities, they form a system of linear equations. Use the data in the problem to write two related equations. Find the slope of each equation by determining the rate of change for each situation. Example Carlos and Carrena work as waiters. Carlos earns $6 per hour plus an average of 20% of his total bills in tips. Carrena earns $5 per hour plus an average of 23% of her total bills in tips. Last Saturday, they worked the same number of hours and had the same total bills. Carlos earned $322 and Carrena earned $357. 6x 1 0.2y x y Using a Graphing Calculator to Solve a Linear System of Equations When the coefficients in a system of equations are not convenient for using substitution or linear combinations, you may need to use a graphing calculator to solve. Follow the steps to graph the system of equations using your graphing calculator and determine the point of intersection. Input the values greater than and less than the solution to interpret the problem situation. Example A Movers: y x y is the total spent on a moving van, and x is the number of miles you need to drive. B Movers: y x The solution is (700, 698). Try 100 miles more or less than the solution. 0.64(600) ; 0.35(600) (800) ; 0.35(800) If you are moving more than 700 miles away, B Movers is the better choice. If you are moving less than 700 miles away, A Movers is the better choice. If you are moving exactly 700 miles, there is no difference between companies. Chapter 12 Summary 683

40 Choosing the Best Method to Solve a Linear System of Equations Use substitution, linear combinations, or a graphing calculator to determine the solution of a linear system of equations. Use substitution to set the equations equal to one another when y is the same for both equations and the equations are in slope-intercept form. Interpret the solution in terms of the problem context. Use linear combinations when the coefficients of like terms are opposites or can be easily made into opposites by multiplication. Use a graphing calculator when the coefficients in a system of equations are complex numbers or are difficult to work with using other methods. Example a. y 5 4x 1 5 3x 1 2y 5 43 Use substitution because one equation is in slope-intercept form. 3x 1 2(4x 1 5) x 1 8x y 5 4(3) x 5 33 y 5 17 x 5 3 The solution is (3, 17). 2x 1 6y 5 14 b. 4x 2 6y 5 10 Use linear combinations because the y-coefficients of both equations are opposites. 2x 1 6y (4) 1 6y x 2 6y y x y 5 6 x 5 4 y 5 1 The solution is (4, 1). 13x 1 5y 5 11 c. 3x 2 16y 5 54 Use a graphing calculator because other methods are not easily used. The solution is (2, 23). 684 Chapter 12 More with Systems of Equations