Solving Linear Systems by Substitution

Transcription

1 COMMON CORE Locker LESSON Common Core Math Standards The student is epected to: COMMON CORE A-REI.C.6 Solve sstems of linear equations eactl... focusing on pairs of linear equations in two variables. Mathematical Practices COMMON CORE 11.2 Solving Linear Sstems b Substitution MP.6 Precision Language Objective Eplain to a partner how to solve a sstem of linear equations b substitution. ENGAGE Essential Question: How can ou solve a sstem of linear equations b using substitution? Solve one equation for one variable and substitute the resulting epression into the other equation. Solve for the value of the other variable in that equation, and then substitute that value into either equation to find the value of the first variable. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss wh a manufacturer might choose to produce either a cheaper version or a more epensive version of a product. Then preview the Lesson Performance Task. Name Class Date 11.2 Solving Linear Sstems b Substitution Essential Question: How can ou solve a sstem of linear equations b using substitution? 1 Eplore Eploring the Substitution Method of Solving Linear Sstems Another method to solve a linear sstem is b using the substitution method. In the sstem of linear equations shown, the value of is given. Use this value of to find the value of and the solution of the sstem. = 2 + = 6 Substitute the value of in the second equation and solve for. + = = 6 = Resource Locker The values of and are known. What is the solution of the sstem? Solution: ( 2, ) Graph the sstem of linear equations. How do our solutions compare? The solutions are the same. Use substitution to find the values of and in this sstem of linear equations. Substitute for in the second equation and solve for. Once ou find the value for, substitute it into either original equation to find the value for. = = 39 Solution: ( 3 12, ) Reflect 1. Discussion For the sstem in Step D, what equation did ou get after substituting for in = 39 and simplifing? 13 = Discussion How could ou check our solution in part D? Graph the sstem or substitute the values of the variables in both of the original equations Module Lesson 2 Name Class Date 11.2 Solving Linear Sstems b Substitution Essential Question: How can ou solve a sstem of linear equations b using substitution? A-REI.C.6 Solve sstems of linear equations eactl... focusing on pairs of linear equations in two variables. 1 Eplore Eploring the Substitution Method of Solving Linear Sstems Resource Another method to solve a linear sstem is b using the substitution method. In the sstem of linear equations shown, the value of is given. Use this value of to find the value of and the solution of the sstem. = 2 + = 6 Substitute the value of in the second equation and solve for. + = 6 + = 6 = The values of and are known. What is the solution of the sstem? Solution: (, ) Graph the sstem of linear equations. How do our solutions compare? Use substitution to find the values of and in this sstem of linear equations. Substitute for in the second equation and solve for. Once ou find the value for, substitute it into either original equation to find the value for. 2 = = 39 Solution: (, ) Reflect The solutions are the same Discussion For the sstem in Step D, what equation did ou get after substituting for in = 39 and simplifing? 13 = 39 Graph the sstem or substitute the values of the variables in both of the original equations. 2. Discussion How could ou check our solution in part D? 2 Module Lesson 2 HARDCOVER Turn to Lesson 11.2 in the hardcover edition. 91 Lesson 11.2

2 Eplain 1 Solving Consistent, Independent Linear Sstems b Substitution The substitution method is used to solve a sstem of equations b solving an equation for one variable and substituting the resulting epression into the other equation. The steps for the substitution method are as shown. 1. Solve one of the equations for one of its variables. 2. Substitute the epression from Step 1 into the other equation and solve for the other variable. 3. Substitute the value from Step 2 into either original equation and solve to find the value of the other variable. Eample = = 7 Solve each sstem of linear equations b substitution. Solve an equation for one variable. 3 + = -3 Select one of the equations. = -3-3 Solve for. Isolate on one side. Substitute the epression for in the other equation and solve (-3-3) = 7 Substitute the epression for = 7 Combine like terms. -5 = 10 Add 3 to both sides. = -2 Divide each side b -5. Substitute the value for into one of the equations and solve for. EXPLORE Eploring the Substitution Method of Solving Linear Sstems INTEGRATE TECHNOLOGY Have students use the graphing tools available in graphing calculators or online to check the solution to a sstem of linear equations. INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP. Make sure that students understand the connection between a sstem of linear equations and its graph. The intersection of the two lines shows the solution of the sstem of equations. 3 (-2) + = -3 Substitute the value of into the first equation = -3 Simplif. = 3 So, (-2, 3) is the solution of the sstem. Check the solution b graphing = = -3 Add 6 to both sides. 3 + = = 7 -intercept: -1 -intercept: - 7_ 2 -intercept: -3 -intercept: 7 The point of intersection is (-2, 3). Module Lesson 2 EXPLAIN 1 Solving Consistent, Independent Linear Sstems b Substitution QUESTIONING STRATEGIES How do ou choose which equation ou solve first and which variable ou solve it for? Eplain. Look for an equation that can easil be solved for one variable, such as an equation in which one variable has a coefficient of 1 or -1. The solution will be the same no matter which equation ou solve first, but this will make the process easier. PROFESSIONAL DEVELOPMENT Learning Progressions In this lesson, students continue their work with sstems of linear equations. Having learned how to solve a sstem b graphing, the now learn how to solve a sstem algebraicall b using the substitution method. The learn how to determine whether a sstem has zero, one, or infinitel man solutions, as well as how to use sstems of linear equations to model real-world situations. As the continue, students will learn other algebraic methods for solving sstems of linear equations, and will learn how to decide which approach is more efficient for a given sstem. Solving Linear Sstems b Substitution 92

3 AVOID COMMON ERRORS Make sure students understand that after ou find the value of one variable, ou must also solve for the other variable. Some students ma consider their work done when the have evaluated one variable. - 3 = 9 B + = 2 Solve an equation for one variable. - 3 = 9 = Substitute the epression for Select one of the equations. Solve for. Isolate on one side. in the other equation and solve. ( ) + = 2 Substitute the epression for = 2 Combine like terms. 9 7 = -7 Subtract from both sides. 7 = -1 Divide each side b. Substitute the value for into one of the equations and solve for ( ) = 9 Substitute the value of into the first equation. + 3 = 9 Simplif. 3 = 6 Subtract from both sides. So, ( 6, -1 ) Reflect is the solution b graphing. Check the solution b graphing. + = = = 9 + = 2 -intercept: 9 -intercept: 2 -intercept: -3 -intercept: The point of intersection is (, ) Eplain how a sstem in which one of the equations if of the form = c, where c is a constant is a special case of the substitution method. There is no need to solve for in terms of because the value of is alread known.. Is it more efficient to solve -2 + = 7 for than for? Eplain. No, because more steps are needed and = _ 2-7_, which is more difficult to substitute 2 than = _ 2 Module Lesson 2 COLLABORATIVE LEARNING Peer-to-Peer Activit Group students in pairs, and give each pair a sstem of linear equations. Have one student solve the first equation for and the other solve the second equation for. Then have both students continue to solve independentl using substitution. Each should arrive at the same solution. Have partners compare their work and discuss which substitution is more efficient. 93 Lesson 11.2

4 Your Turn 5. Solve the sstem of linear equations b substitution. 3 + = 1 = = (-3 + 1) = -2 = 3 Eplain 2 Solving Special Linear Sstems b Substitution You can use the substitution method for sstems of linear equations that have infinitel man solutions and for sstems that have no solutions. Eample 2 Solve each sstem of linear equations b substitution. + = - - = 6 Solve + = for. = - + Substitute the resulting epression into the other equation and solve. - (- + ) - = 6 Substitute. - = 6 Simplif. The resulting equation is false, so the sstem has no solutions. - 3 = 6-12 = 2 Solve - 3 = 6 for. = Substitute the resulting epression into the other equation and solve. ( ) - 12 = 2 Substitute. 2 = 2 Simplif. The resulting equation is true, so the infinitel man solutions sstem has. Reflect The graph shows that the lines are parallel and do not intersect. The graphs are the same line, infinitel man solutions so the sstem has. 6. Provide two possible solutions of the sstem in Eample 2B. How are all the solutions of this sstem related to one another? Sample solutions: (0, -2) and (6, 0) ; all the solutions of this sstem are points on the line (3) + = 1 = 5 The solution of the sstem is (3, 5) = = - 3 = = 2 - Module 11 9 Lesson 2 EXPLAIN 2 Solving Special Linear Sstems b Substitution QUESTIONING STRATEGIES When solving a sstem of linear equations b substitution, how can ou tell if the sstem has no solution or infinitel man solutions? If it has no solutions, the solution process will result in an equation that is false. If it has infinitel man solutions, the solution process will result in an equation that is alwas true. How does the graph of a sstem of linear equations tell ou it has no solution or infinitel man solutions? When the equations represent two parallel lines, the lines do not intersect, so there is no solution. When both equations represent the same line, the sstem has infinitel man solutions. AVOID COMMON ERRORS Make sure students understand that when ou substitute an epression for a variable, the epression should be placed inside parentheses. Remind students to follow the order of operations and to appl the Distributive Propert correctl when dealing with epressions inside parentheses. INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 You can use algebra tiles to model and solve some sstems of linear equations. Solve for one variable using the first equation, then model the second equation. Solving Linear Sstems b Substitution 9

5 EXPLAIN 3 Solving Linear Sstem Models b Substitution AVOID COMMON ERRORS Some students ma struggle with solving b substitution because the automaticall start b solving the first equation for. Encourage them to look at both equations and check whether an of the variables has a coefficient of 1 or -1. Then have students solve for that variable first. QUESTIONING STRATEGIES Is it more accurate to check our solution b graphing or b substituting back into the original equations? Eplain. Substituting, because if the solution does not consist of integers, graphing ma not give an accurate check. INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP.5 Some real-world problems, especiall those involving mone, ma have sstems of equations with decimal coefficients. It is far easier to use a graphing calculator to check the solution than it is to draw the graph b hand. Your Turn Solve each sstem of linear equations b substitution = -2-7 = 1 Eplain = = 1 Solving Linear Sstem Models b Substitution You can use a sstem of linear equations to model real-world situations. Eample 3 = (7 + 1) + 1 = -2 Solve each real-world situation b using the substitution method. Fitness center A has a $60 enrollment fee and costs $35 per month. Fitness center B has no enrollment fee and costs $5 per month. Let t represent the total cost in dollars and m represent the number of months. The sstem of equations t = m can be used t = 5m to represent this situation. In how man months will both fitness centers cost the same? What will the cost be? m = 5m Substitute m for t in the second equation. t = 5m -2 = -2 infinitel man solutions 60 = 10m Subtract 35m from each side. 6 = m Divide each side b 10. Use one of the original equations. = 5 (6) = 270 Substitute 6 for m. (6, 270) Write the solution as an ordered pair. Both fitness centers will cost $270 after 6 months. = (3 + 12) = 1 no solutions High-speed Internet provider A has a $100 setup fee and costs $65 per month. High-speed internet provider B has a setup fee of $30 and costs $70 per month. Let t represent the total amount paid in dollars and m represent the number of months. The sstem of equations t = m t = m can be used to represent this situation. In how man months will both providers cost the same? What will that cost be? m -2 = 1 = m Substitute m for t in the second equation. 100 = m Subtract 65 m from each side. 70 = 5 m Subtract 30 from each side. 1 = m Divide each side b 5. Module Lesson 2 DIFFERENTIATE INSTRUCTION Graphic Organizer Have students show the steps for solving a sstem of equations b substitution. Solving Sstems of Equations b Substitution Step 1 Solve for one variable in one equation. Step 2 Substitute the resulting epression into the other equation. Step 3 Solve that equation to get the value of the other variable. Step Substitute that value into one of the original equations and solve. Step 5 Write the values from Steps 3 and in an ordered pair (, ). Step 6 Check the solution b substituting into both equations or b graphing. 95 Lesson 11.2

6 t = m 1 Use one of the original equations. 1 t = ( ) Substitute for m. t = ( 1, 1010 ) Write the sulotion as an ordered pair. Both Internet providers will cost $ 1010 after 1 months. Reflect 9. If the variables in a real-world situation represent the number of months and cost, wh must the values of the variables be greater than or equal to zero? The values of the variables must be greater than or equal to zero because the total cost and the number of months cannot be negative. Your Turn A boat travels at a rate of 1 kilometers per hour from its port. A second boat is 3 kilometers behind the first boat when it starts traveling in the same direction at a rate of 22 kilometers per hour to the same port. Let d represent the distance the boats are from the port in kilometers and t represent the amount of time in hours. The sstem of equations d = 1t + 3 can be used to represent this situation. How man d = 22t hours will it take for the second boat to catch up to the first boat? How far will the boats be from their port? Use the substitution method to solve this real-world application. 1t + 3 = 22t d = 22t.5 = t d = 22 (.5) = 17 The second boat will catch up in.5 hours, and the will be 17 km from their port. Elaborate ELABORATE QUESTIONING STRATEGIES Wh can ou substitute the value of one variable into either of the original equations to find the value of the other variable? If there is a solution to the sstem of equations, the values of the variables will satisf both equations. SUMMARIZE THE LESSON How do ou know if our solution to a sstem of linear equations is correct? You can verif our solution b graphing the equations. This allows ou to verif that the number of solutions is correct b seeing whether the lines appear to be the same line, two parallel lines, or two lines that intersect at one point. If the lines intersect at one point, ou can also substitute the solution back into the original equations to verif our solution. 11. When given a sstem of linear equations, how do ou decide which variable to solve for first? Use a variable that has a coefficient of 1 or -1. If no variables have a coefficient of 1 or -1, look for a variable that will result in the simplest epression. 12. How can ou check a solution for a sstem of equations without graphing? Substitute the solution into each equation and determine whether all of the equations in the sstem are true. 13. Essential Question-Check-In Eplain how ou can solve a sstem of linear equations b substitution. Solve one equation for one variable and use the result to substitute into the other equation. Solve for the value of the other variable. Then substitute that value into either equation to find the value of the first variable. Module Lesson 2 LANGUAGE SUPPORT Connect Vocabular Remind students that the substitution method involves substituting an epression from one equation into the other equation. Eplain that to substitute means to replace. Note that the word substitute can be used as a noun or an adjective: a substitute (noun) in sports replaces the original plaer, and a substitute (adjective) teacher replaces the regular teacher. Emphasize that epressions used for substitution in math must alwas be equal in value to the epression the are replacing. Solving Linear Sstems b Substitution 96

7 EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore Eploring the Substitution Method of Solving Linear Sstems Eample 1 Solving Consistent, Independent Linear Sstems b Substitution Eample 2 Solving Special Linear Sstems b Substitution Eample 3 Solving Linear Sstem Models b Substitution Practice Eercise 1 Eercises 2 7, Eercises 13, 20 Eercises 1 19, INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Students can check their solutions for correctness b substituting the values into the original equations and verifing that both solutions make both equations true. 1. In the sstem of linear equations shown, the value of is given. Use this value of to find the value of and the solution of the sstem. = = a. What is the solution of the sstem? b. Graph the sstem of linear equations. How do the solutions compare? The solution is (, 12). 16 Solve each sstem of linear equations b substitution Evaluate: Homework and Practice 5 + = 2 + = 5 = (-5 + ) = = 5-3 = -3 = 1 5 (1) + = 5 + = = 3 The solution is (1, 3). + 7 = = = (-7-11) - 5 = = 9 = -1 = (-2) = = -11 = 3 The solution is (3, -2) = = = = = (3 + 10) + 5 = = -22 = -32 = (-) = = 10 = -2 The solution is (-2, -). 2 = = (-3 + ) - 5 = = -1-1 = -2 = (3) = = 16 2 = -2 = -1 The solution is (-1, 3) The solutions are the same. 5-3 = = = = 3 Online Homework Hints and Help Etra Practice = ( - 19) = = 22-7 = -35 = 5 - (5) + = = -19 = 1 The solution is (5, 1). 3 = = (2 + 1) = = 2 11 = 22 = 2-6 (2) + 3 = = 3 3 = 15 = 5 The solution is (2, 5). Module Lesson 2 Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 1 2 Skills/Concepts MP. Modeling Recall of Information MP.2 Reasoning Skills/Concepts MP. Modeling 19 3 Strategic Thinking MP. Modeling 20 2 Skills/Concepts MP.2 Reasoning 21 2 Skills/Concepts MP. Modeling 97 Lesson 11.2

8 Solve each sstem of linear equations b substitution = = 12 = (- + 3) - = = 12 There is no solution. 5 - = = 32 - = = (5-1) = = 32 There is no solution = = 5 = ( + 5) = = -15 There are infinitel man solutions = = 2 = - 3_ ( - 3_ 2-6 ) - 6 = 2 2 = 2 There are infinitel man solutions. Solve each real-world situation b using the substitution method. 1. The number of DVDs sold at a store in a month was 920 and the number of DVDs sold decreased b 12 per month. The number of Blu-ra discs sold in the same store in the same month was 502 and the number of Blu-ra discs sold increased b 26 per month. Let d represent the number of discs sold and t represent the time in months. The sstem of equations d = t can be d = t = = -51 = ( + 17) + 2 = = -51 There are infinitel man solutions. 3 + = = 3 = = - _ ( - _ ) + = 72 = There is no solution. used to represent this situation. If this trend continues, in how man months will the number of DVDs sold equal the number of Blu-ra discs sold? How man of each is sold in that month? t = t d = t 1 = 3t d = (11) 11 = t d = 7 There will be 7 DVDs and 7 Blu-Ra discs sold per month in 11 months. 15. One smartphone plan costs $30 per month for talk and messaging and $ per gigabte of data used each month. A second smartphone plan costs $60 per month for talk and messaging and $3 per gigabte of data used each month. Let c represent the total cost in dollars and d represent the amount of data used in gigabtes. The sstem of equations c = 30 + d can be used to represent this situation. How man c = d gigabtes would have to be used for the plans to cost the same? What would that cost be? 30 + d = d c = 30 + (6) 5d = 30 c = 7 d = 6 Both plans would cost $7 if 6 gigabtes of data are used. Image Credits: StockPhotosArt/Shutterstock MODELING Some students ma have difficult using the substitution method. Suggest to them that the graph the sstem first, and then use the graph to guide and check their work as the use substitution. INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Remind students that when using the substitution method to solve a sstem, it does not matter which variable ou solve for first. Demonstrate that whether ou solve for first or first, ou will obtain the same solution. Therefore, ou can choose to solve in whichever order is easier. If possible, solve for the variable that has a coefficient of 1 or -1. Module 11 9 Lesson 2 Eercise Depth of Knowledge (D.O.K.) COMMON CORE Mathematical Practices 22 3 Strategic Thinking MP. Modeling 23 3 Strategic Thinking MP.6 Precision Strategic Thinking MP.3 Logic Solving Linear Sstems b Substitution 9

9 AVOID COMMON ERRORS Students often think the have solved a sstem of equations after finding the value of onl one variable. Remind them that the solution is an ordered pair. 16. A movie theater sells popcorn and fountain drinks. Brett bus 1 popcorn bucket and 3 fountain drinks for his famil, and pas a total of $9.50. Sarah bus 3 popcorn buckets and fountain drinks for her famil, and pas a total of $ If p represents the number of popcorn buckets and d represents the number of drinks, then the sstem of equations 9.50 = p + 3d can be used to represent this situation. Find the = 3p + d cost of a popcorn bucket and the cost of a fountain drink d = p 9.50 = p + 3 (1.75) = 3 (9.50-3d) + d 9.50 = p = d.25 = p The cost of a bucket of popcorn is $.25 and the cost of a fountain soda is $ Jen is riding her biccle on a trail at the rate of 0.3 kilometer per minute. Michelle is 11.2 kilometers behind Jen when she starts traveling on the same trail at a rate of 0. kilometer per minute. Let d represent the distance in kilometers the bicclists are from the start of the trail and t represent the time in minutes. d = 0.3t The sstem of equations can be used to represent this situation. How man minutes d = 0.t will it take Michelle to catch up to Jen? How far will the be from the start of the trail? Use the substitution method to solve this real-world application. 0.3t = 0.t d = 0.t 0 = t = 0. (0) = 35.2 Michelle will catch up in 0 minutes, and the will be 35.2 km from the start. Image Credits: StockPhotosArt/Shutterstock 1. Geometr The length of a rectangular room is 5 feet more than its width. The perimeter of the room is 66 feet. Let L represent the length of the room and W represent the width in feet. The sstem of equations L = W + 5 can be used to represent this situation. What are the room s dimensions? 66 = 2L + 2W 66 = 2 (W + 5) + 2W L = W = W L = = W L = 19 The room has a width of 1 feet and a length of 19 feet. 19. A cable television provider has a $55 setup fee and charges $2 per month, while a satellite television provider has a $160 setup fee and charges $67 per month. Let c represent the total cost in dollars and t represent the amount of time in months. The sstem of equations c = t can be used to represent c = t this situation. a. In how man months will both providers cost the same? What will that cost be? t = t c = t 15t = 105 c = (7) t = 7 = 629 Both providers will cost $629 in 7 months. b. If ou plan to move in 12 months, which provider would be less epensive? Eplain. Satellite would be less epensive because it costs less per month than cable and 12 months is after 7 months. Module Lesson 2 99 Lesson 11.2

10 20. Determine whether each of the following sstems of equations have one solution, infinitel man solutions, or no solution. Select the correct answer for each lettered part. a. + = 5 none b. + = 7 one -6-6 = = 23 c. 3 + = = 12 e = = -3 d = = Finance Adrienne invested a total of $1900 in two simple-interest mone market accounts. Account A paid 3% annual interest and account B paid 5% annual interest. The total amount of interest she earned after one ear was $3. If a represents the amount invested in dollars in account A and b represents the amount invested in dollars in account B, the sstem of equations a + b = 1900 can represent 0.03a b = 3 this situation. How much did Adrienne invest in each account? a = -b none infinitel man 0.03 (-b ) b = b = 26 a + b = 1900 a + (1300) = 1900 a = 600 b = 1300 Adrienne invested $600 in account A and $1300 in account B. one VISUAL CUES After isolating one variable in one equation, some students ma find it helpful to highlight the variable with a colored pencil, and then highlight the same variable in the other equation. This will help them remember where in the other equation to substitute the epression for that variable. H.O.T. Focus on Higher Order Thinking 22. Real-World Application The Sullivans are deciding between two landscaping companies. Evergreen charges a $79 startup fee and $39 per month. Eco Solutions charges a $25 startup fee and $5 per month. Let c represent the total cost in dollars and t represent the time in months. The sstem of equations c = 39t + 79 can be used to represent this c = 5t + 25 situation. a. In how man months will both landscaping services cost the same? What will that cost be? 39t + 79 = 5t + 25 c = 5t = 6t c = 39 (9) = t = 30 Both will cost $30 in 9 months. b. Which landscaping service will be less epensive in the long term? Eplain. Evergreen will be less epensive than Eco Solutions in the long term. The will cost the same after 9 months but the rate of change for Evergreen is less than the rate of change for Eco Solutions. Image Credits: Bill Hustace/Corbis Module Lesson 2 Solving Linear Sstems b Substitution 500

11 JOURNAL Have students write a journal entr that summarizes how to solve a sstem of equations b substitution. Students should mention how to decide which equation to use for the substitution. 23. Multiple Representations For the first equation in the sstem of linear equations below, write an equivalent equation without denominators. Then solve the sstem. _ 5 + _ 3 = 6-2 = 15 ( _ 5 + _ 3 ) = = 90-2 = = (2 + ) + 5 = = = = 66 = 6-2 (6) = - 12 = = 20 The solution is (20, 6). 2. Conjecture Is it possible for a sstem of three linear equations to have one solution? If so, give an eample. Yes; the solution is an ordered pair that is a solution of each of the equations. For eample, the solution of the sstem containing the equations 3 - = 5, + = 3, and = 2 is (2, 1). 25. Conjecture Is it possible to use substitution to solve a sstem of linear equations if one equation represents a horizontal line and the other equation represents a vertical line? Eplain. No, the equation of a horizontal line is in the form = a and the equation of a vertical line is in the form = b. The horizontal line equation has no -term and the vertical line equation has no -term. Module Lesson Lesson 11.2

12 Lesson Performance Task A compan breaks even from the production and sale of a product if the total revenue equals the total cost. Suppose an electronics compan is considering producing two tpes of smartphones. To produce smartphone A, the initial cost is $20,000 and each phone costs $150 to produce. The compan will sell smartphone A at $200. Let C(a) represent the total cost in dollars of producing a units of smartphone A. Let R(a) represent the total revenue, or mone the compan takes in due to selling a units of smartphone A. The sstem of equations C (a) = 20, a can be used to represent the situation for phone A. R (a) = 200a To produce smartphone B, the initial cost is $,000 and each phone costs $200 to produce. The compan will sell smartphone B at $20. Let C(b) represent the total cost in dollars of producing b units of smartphone B and R(b) represent the total revenue from QUESTIONING STRATEGIES How is profit determined? Profit = total revenue - total cost = R - C When total revenue equals total cost, what is the profit? What is this situation called? The profit is $0; this is called the break-even point. The break-even points in the Lesson Performance Task are when C(a) = R(a) for smartphone A and C(b) = R(b) for smartphone B. selling b units of smartphone B. The sstem of equations C (b) =, b can be R (b) = 20b used to represent the situation for phone B. Solve each sstem of equations and interpret the solutions. Then determine whether the compan should invest in producing smartphone A or smartphone B. Justif our answer. Smartphone A: 200a = 20, a 50a = 20,000 a = 00 R (a) = 200a = 200 (00) = 0,000 The compan will break even selling 00 units of smartphone A for a total $0,000. Smartphone B: 20b =, b 0b =,000 b = 550 R (b) = 20b = 20 (550) = 15,000 The compan will break even selling 550 units of smartphone B for a total of $15,000. Some students ma sa that the compan should invest in producing smartphone A because the initial cost of producing smartphone A is less than that of producing smartphone B and fewer units of smartphone A would need to be sold for the compan to break even. Other students ma argue that the compan should consider other factors, such as increasing the sale price of smartphone B or looking for was to cut the initial cost of production. Module Lesson 2 Image Credits: Bill Hustace/Corbis INTEGRATE MATHEMATICAL PRACTICES Focus on Technolog MP. To check their solutions for smartphone A, have students use graphing calculators to graph = 20, and = 200 on the same coordinate plane. Then the can go to the CALC menu and select the intersect feature to find the coordinates of the point of intersection. Students can use the same procedure with the equations =, and = 20 to check their answers for smartphone B. EXTENSION ACTIVITY Man companies sell accessories for smartphones. Have students research the different tpes of accessories sold and make conjectures about how a compan might use a sstem of equations to find the break-even cost in selling these accessories. Scoring Rubric 2 points: Student correctl solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not full solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Solving Linear Sstems b Substitution 502