11.1 Solving Linear Systems by Graphing

Transcription

1 Name Class Date 11.1 Solving Linear Sstems b Graphing Essential Question: How can ou find the solution of a sstem of linear equations b graphing? Resource Locker Eplore Tpes of Sstems of Linear Equations A sstem of linear equations, also called a linear sstem, consists of two or more linear equations that have the same variables. A solution of a sstem of linear equations with two variables is an ordered pair that satisfies all of the equations in the sstem. A Describe the relationship between the two lines in Graph A. Graph A B What do ou know about ever point on the graph on a linear equation? C How man solutions does a sstem of two equations have if the graphs of the two equations intersect at eactl one point? Houghton Mifflin Harcourt Publishing Compan D E Describe the relationship between the two lines that coincide in Graph B. How man solutions does a sstem of two equations have if the graphs of the two equations intersect at infinitel man points? Graph B Module Lesson 1

2 Describe the relationship between the two lines in Graph C. Graph C How man solutions does a sstem of two equations have if the graphs of the two equations do not intersect? Reflect 1. Discussion Eplain wh the solution of a sstem of two equations is represented b an point where the two graphs intersect. Eplain 1 Solving Consistent, Independent Linear Sstems b Graphing A consistent sstem is a sstem with at least one solution. Consistent sstems can be either independent or dependent. An independent sstem has eactl one solution. The graph of an independent sstem consists of two lines that intersect at eactl one point. A dependent sstem has infinitel man solutions. The graph of a dependent sstem consists of two coincident lines, or the same line. A sstem that has no solution is an inconsistent sstem. Eample 1 Solve the sstem of linear equations b graphing. Check our answer. + = 6 A - + = 3 Find the intercepts for each equation, plus a third point for a check. Then graph. + = = 3 -intercept: 3 -intercept: 3 -intercept: 6 -intercept: 3 third point: (-1, ) third point: (3, 6) The two lines appear to intersect at (1, ). Check. + = = 3 (1) + 6 -(1) = = 6-0 Houghton Mifflin Harcourt Publishing Compan 6 = 6 3 = 3 The point satisfies both equations, so the solution is (1, ). Module 11 0 Lesson 1

3 B = = 1 Find the intercepts for each equation, plus a third point for a check. Then graph. = = 1 -intercept: -intercept: -intercept: -intercept: third point: (3, ) third point: (1, ) The two lines appear to intersect at. Check. = - + = ( ) - + ( ) 6 = = 6 The point satisfies both equations, so the solution is. Reflect. How do ou know that the sstems of equations are consistent? How do ou know that the are independent? Your Turn Solve the sstem of linear equations b graphing. Check our answer. 3. = + =. = + + = 6 Houghton Mifflin Harcourt Publishing Compan Module 11 1 Lesson 1

4 Eplain Solving Special Linear Sstems b Graphing Eample Solve the special sstem of equations b graphing and identif the sstem. = - A - + = Find the intercepts for each equation, plus a third point for a check. = = -intercept: 1 -intercept: - -intercept: - -intercept: third point: (, ) third point: (, ) The two lines don t intersect, so there is no solution. The two lines have the same slope and different -intercepts so the will never intersect. This is an inconsistent sstem. = 3-3 B -3 + = -3 Find the intercepts for each equation, plus a third point for a check. = = -3 -intercept: -intercept: -intercept: -intercept: third point: (, ) third point: (, ) The two lines coincide, so there are solutions. The have the same slope and -intercept; therefore, the are line(s) / equation(s). This is a and sstem. \ Your Turn Solve the special sstem of linear equations b graphing. Check our answer. 5. = = = _ _ 3 + = Houghton Mifflin Harcourt Publishing Compan Module 11 Lesson 1

5 Eplain 3 Estimating Solutions of Linear Sstems b Graphing You can estimate the solution of a linear sstem of equations b graphing the sstem and finding the approimate coordinates of the intersection point. Eample 3 Estimate the solution of the linear sstem b graphing. - 3 = 3 A -5 + = 10 Graph the equations using a graphing calculator. Y1 = (3 X)/( 3) and Y = (10 + 5X)/ Find the point of intersection. The two lines appear to intersect at about (., 1.9). 6 - = 3 B = + 6 Graph each equation b finding intercepts. The two lines appear to intersect at about. Check to see if makes both equations true. 6 - = 3 = ( ) 3 ( ) ( ) + The point does not satisf both equations, but the results are close. So, is an approimate solution. Houghton Mifflin Harcourt Publishing Compan Your Turn Estimate the solution of the linear sstem of equations b graphing. 7. = = = -9 = 1_ Module 11 3 Lesson 1

6 Eplain Interpreting Graphs of Linear Sstems to Solve Problems You can solve problems with real-world contet b graphing the equations that model the problem and finding a common point. Eample Rock and Bowl charges $.75 per game plus $3 for shoe rental. Super Bowling charges $.5 per game and $3.50 for shoe rental. For how man games will the cost to bowl be approimatel the same at both places? What is that cost? Analze Information Identif the important information. Rock and Bowl charges $ per game plus $ for shoe rental. Super Bowling charges $ per game and $ for shoe rental. The answer is the number of games plaed for which the total cost is approimatel the same at both bowling alles. Formulate a Plan Write a sstem of linear equations, where each equation represents the price at each bowling alle. Solve Graph = and = The lines appear to intersect at places will be the same for cost will be. Justif and Evaluate Reflect Check using both equations.. So, the cost at both game(s) bowled and that.75 ( ) + 3 =.5 ( ) = 9. Which bowling alle costs more if ou bowl more than 1 game? Eplain how ou can tell b looking at the graph. Cost ($) Games Houghton Mifflin Harcourt Publishing Compan Image Credits: Ilene MacDonald/Alam Module 11 Lesson 1

7 Your Turn 10. Video club A charges $10 for membership and $ per movie rental. Video club B charges $15 for membership and $3 per movie rental. For how man movie rentals will the cost be the same at both video clubs? What is that cost? Write a sstem and solve b graphing. Cost ($) Number of movies Elaborate 11. When a sstem of linear equations is graphed, how is the graph of each equation related to the solutions of that equation? 1. Essential Question Check-In How does graphing help ou solve a sstem of linear equations? Evaluate: Homework and Practice Houghton Mifflin Harcourt Publishing Compan Image Credits: Ilene MacDonald/Alam 1. Is the following statement correct? Eplain. A sstem of two equations has no solution if the graphs of the two equations are coincident lines. Solve the sstem of linear equations b graphing. Check our answer.. = - + = 1 3. = - 1_ = - _ 3 Online Homework Hints and Help Etra Practice Module 11 5 Lesson 1

8 . = = _ = 3 - = = - + = 1_ + = _ 3 + = = 1_. - = _ = = = = = = Houghton Mifflin Harcourt Publishing Compan - - Module 11 6 Lesson 1

9 = - 3_ _ = _ + - = = Estimate the solution of the linear sstem of equations b graphing. 1. = = = 1-35 = Houghton Mifflin Harcourt Publishing Compan _ = = = = - _ Module 11 7 Lesson 1

10 Solve b graphing. Give an approimate solution if necessar. 1. Wren and Jenni are reading the same book. Wren is on page 1 and reads 3 pages ever night. Jenni is on page 7 and reads pages ever night. After how man nights will the have read the same number of pages? How man pages will that be? 19. Rust burns 6 calories per minute swimming and 10 calories per minute jogging. In the morning, Rust burns 175 calories walking and swims for minutes. In the afternoon, Rust will jog for minutes. How man minutes must he jog to burn at least as man calories in the afternoon as he did in the morning? Round our answer up to the net whole number of minutes. 0. A gm membership at one gm costs $10 ever month plus a onetime membership fee of $15, and a gm membership at another gm costs $ ever month plus a one-time $0 membership fee. After about how man months will the gm memberships cost the same amount? 1. Malor is putting mone in two savings accounts. Account A started with $150 and Account B started with $300. Malor deposits $16 in Account A and $1 in Account B each month. In how man months will Account A have a balance at least as great as Account B? What will that balance be? Houghton Mifflin Harcourt Publishing Compan Module 11 Lesson 1

11 . Critical Thinking Write sometimes, alwas, or never to complete the following statements. a. If the equations in a sstem of linear equations have the same slope, there are infinitel man solutions for the sstem. b. If the equations in a sstem of linear equations have different slopes, there is one solution for the sstem. c. If the equations in a sstem of linear equations have the same slope and a different -intercept, there is an solution for the sstem. H.O.T. Focus on Higher Order Thinking 3. Critique Reasoning Brad classifies the sstem below as inconsistent because the equations have the same -intercept. What is his error? = - = -. Eplain the Error Alea solved the sstem + = = - b graphing and estimated the solution to be about (1.5, 0.6). What is her error? What is the correct answer? Houghton Mifflin Harcourt Publishing Compan 5. Represent Real-World Problems Cora ran 3 miles last week and will run 7 miles per week from now on. Hana ran 9 miles last week and will run miles per week from now on. The sstem of linear equations = can be used to represent = + 9 this situation. Eplain what and represent in the equations. After how man weeks will Cora and Hana have run the same number of miles? How man miles? Solve b graphing. Module 11 9 Lesson 1

12 Lesson Performance Task A boat takes 7.5 hours to make a 60-mile trip upstream and 6 hours on the 60-mile return trip. Let v be the speed of the boat in still water and c be the speed of the current. The upstream speed of the boat is v - c and the downstream boat speed is v + c. a. Use the distance formula to write a sstem of equations relating boat speed and time to distance, one equation for the upstream part of the trip and one for the downstream part. b. Graph the sstem to find the speed of the boat in still water and the speed of the current. 10 c v c. How long would it take the boat to travel the 60 miles if there were no current? Houghton Mifflin Harcourt Publishing Compan Module Lesson 1

13 Name Class Date 11. Solving Linear Sstems b Substitution Essential Question: How can ou solve a sstem of linear equations b using substitution? Resource Locker 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. = + = 6 A Substitute the value of in the second equation and solve for. B The values of and are known. What is the solution of the sstem? + = 6 + = 6 Solution: (, ) = Houghton Mifflin Harcourt Publishing Compan C D 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. = 5 + = 39 Solution: (, ) Reflect Discussion For the sstem in Step D, what equation did ou get after substituting for in 5 + = 39 and simplifing? - -. Discussion How could ou check our solution in part D? Module Lesson

14 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.. Substitute the epression from Step 1 into the other equation and solve for the other variable. 3. Substitute the value from Step into either original equation and solve to find the value of the other variable. Eample 1 Solve each sstem of linear equations b substitution. 3 + = -3 A - + = 7 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. = - Divide each side b -5. Substitute the value for into one of the equations and solve for. 3 (-) + = -3 Substitute the value of into the first equation = -3 Simplif. = 3 So, (-, 3) is the solution of the sstem. Check the solution b graphing = 3 + = -3 Add 6 to both sides. 3 + = = 7 -intercept: -1 -intercept: - 7_ -intercept: -3 -intercept: 7 The point of intersection is (-, 3). Houghton Mifflin Harcourt Publishing Compan Module 11 9 Lesson

15 B - 3 = 9 + = Solve an equation for one variable. - 3 = 9 = Select one of the equations. Solve for. Isolate on one side. Substitute the epression for in the other equation and solve. ( ) + = Substitute the epression for. = Combine like terms. = Subtract from both sides. = Divide each side b. Substitute the value for into one of the equations and solve for. - 3 ( ) = 9 Substitute the value of into the first equation. = 9 Simplif. = Subtract from both sides. So, (, ) is the solution b graphing. Check the solution b graphing. Houghton Mifflin Harcourt Publishing Compan - Reflect = 9 + = -intercept: -intercept: -intercept: -intercept: The point of intersection is (, ). 3. 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.. Is it more efficient to solve - + = 7 for than for? Eplain. Module Lesson

16 Your Turn 5. Solve the sstem of linear equations b substitution. 3 + = 1-6 = - Eplain 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 Solve each sstem of linear equations b substitution. + = A - - = 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 = The graph shows that the lines are parallel and do not intersect. + = - 3 = 6 B - 1 = Solve - 3 = 6 for. = Substitute the resulting epression into the other equation and solve. Reflect ( ) - 1 = Substitute. The resulting equation is = Simplif., so the sstem has The graphs are, so the sstem has. Houghton Mifflin Harcourt Publishing Compan 6. Provide two possible solutions of the sstem in Eample B. How are all the solutions of this sstem related to one another? Module 11 9 Lesson

17 Your Turn Solve each sstem of linear equations b substitution = = = = 1 Eplain 3 Solving Linear Sstem Models b Substitution You can use a sstem of linear equations to model real-world situations. Eample 3 Solve each real-world situation b using the substitution method. A 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? Houghton Mifflin Harcourt Publishing Compan B m = 5m Substitute m for t in the second equation. t = 5m 60 = 10m Subtract 35m from each side. 6 = m Divide each side b 10. Use one of the original equations. = 5 (6) = 70 Substitute 6 for m. (6, 70) Write the solution as an ordered pair. Both fitness centers will cost $70 after 6 months. 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 Substitute for t in the second equation. 100 = Subtract m from each side. = m Subtract from each side. = m Divide each side b. Module Lesson

18 t = m Use one of the original equations. t = ( ) Substitute for m. t = (, ) Write the sulotion as an ordered pair. Both Internet providers will cost $ after 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? Your Turn 10. 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 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 = t 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. Elaborate 11. When given a sstem of linear equations, how do ou decide which variable to solve for first? 1. How can ou check a solution for a sstem of equations without graphing? 13. Essential Question-Check-In Eplain how ou can solve a sstem of linear equations b substitution. Houghton Mifflin Harcourt Publishing Compan Module Lesson

19 Evaluate: Homework and Practice 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. = 1 - = a. What is the solution of the sstem? b. Graph the sstem of linear equations. How do the solutions compare? 16 Online Homework Hints and Help Etra Practice Solve each sstem of linear equations b substitution = + = = = = - + = -19 Houghton Mifflin Harcourt Publishing Compan = = = = = = 3 Module Lesson

20 Solve each sstem of linear equations b substitution.. + = = = = = = = = = = = = Solve each real-world situation b using the substitution method. 1. The number of DVDs sold at a store in a month was 90 and the number of DVDs sold decreased b 1 per month. The number of Blu-ra discs sold in the same store in the same month was 50 and the number of Blu-ra discs sold increased b 6 per month. Let d represent the number of discs sold and t represent the time in months. The sstem of equations d = 90-1t can be d = t 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? 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 = 30 + d c 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? Houghton Mifflin Harcourt Publishing Compan Image Credits: StockPhotosArt/Shutterstock Module 11 9 Lesson

21 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. 17. Jen is riding her biccle on a trail at the rate of 0.3 kilometer per minute. Michelle is 11. 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. The sstem of equations = 0.3t d 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. 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 = L + W Houghton Mifflin Harcourt Publishing Compan Image Credits: StockPhotosArt/Shutterstock 19. A cable television provider has a $55 setup fee and charges $ 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 = 55 + t c 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? b. If ou plan to move in 1 months, which provider would be less epensive? Eplain. Module Lesson

22 0. 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. + = = 30 b. + = = 3 c. 3 + = = 1 e = = -3 d. + 5 = = 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? H.O.T. Focus on Higher Order Thinking. 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 $5 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 = 39t + 79 c can be used to represent this c = 5t + 5 situation. a. In how man months will both landscaping services cost the same? What will that cost be? b. Which landscaping service will be less epensive in the long term? Eplain. Houghton Mifflin Harcourt Publishing Compan Image Credits: Bill Hustace/Corbis Module Lesson

23 3. 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 - =. Conjecture Is it possible for a sstem of three linear equations to have one solution? If so, give an eample. Houghton Mifflin Harcourt Publishing Compan 5. 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. Module Lesson

24 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 $0,000 and each phone costs $150 to produce. The compan will sell smartphone A at $00. 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 0, a C (a) = can be used to represent the situation for phone A. R (a) = 00a To produce smartphone B, the initial cost is $,000 and each phone costs $00 to produce. The compan will sell smartphone B at $0. Let C(b) represent the total cost in dollars of producing b units of smartphone B and R(b) represent the total revenue from selling b units of smartphone B. The sstem of equations, b C (b) = can be R (b) = 0b 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. Houghton Mifflin Harcourt Publishing Compan Image Credits: Bill Hustace/Corbis Module Lesson

25 Name Class Date 11.3 Solving Linear Sstems b Adding or Subtracting Essential Question: How can ou solve a sstem of linear equations b adding and subtracting? Resource Locker Eplore Eploring the Effects of Adding Equations Sstems of equations can be solved b graphing, substitution, or b a third method, called elimination. A Look at the sstem of linear equations. - = = 5 What do ou notice about the coefficients of the -terms? B What is the sum of - and? How do ou know? C Find the sum of the two equations b combining like terms. D Use the equation from Step C to find the value of. - = = +5 + = = Houghton Mifflin Harcourt Publishing Compan E Use the value of to find the value of. What is the solution of the sstem? = Solution: Reflect 1. Discussion How do ou know that when both sides of the two equations were added, the resulting sums were equal?. Discussion How could ou check our solution? Module Lesson 3

26 Eplain 1 Solving Linear Sstems b Adding or Subtracting The elimination method is a method used to solve sstems of equations in which one variable is eliminated b adding or subtracting two equations in the sstem. Steps in the Elimination Method 1. Add or subtract the equations to eliminate one variable, and then solve for the other variable.. Substitute the value into either original equation to find the value of the eliminated variable. 3. Write the solution as an ordered pair. Eample 1 Solve each sstem of linear equations using the indicated method. Check our answer b graphing. A Solve the sstem of linear equations b adding. - = 1 + = Add the equations. - = 1 + = = 0 5 = 0 = Substitute the value of into one of the equations and solve for. + = + = = = Write the solution as an ordered pair. (, ) + = = 1 B Check the solution b graphing. Solve the sstem of linear equations b subtracting. + 6 = 6 Substitute the value of into one of the equations and solve for. - = - Subtract the equations. + 6 = 6 - = - = Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

27 Write the solution as an ordered pair Check the solution b graphing. Reflect 3. Can the sstem in part A be solved b subtracting one of the original equations from the other? Wh or wh not?. In part B, what would happen if ou added the original equations instead of subtracting? Houghton Mifflin Harcourt Publishing Compan Your Turn Solve each sstem of linear equations b adding or subtracting = = = = -1 Module Lesson 3

28 Eplain Solving Special Linear Sstems b Adding or Subtracting Eample Solve each sstem of linear equations b adding or subtracting. - - = A + = - Add the equations. - - = + + = = 0 0 = 0 The resulting equation is true, so the sstem has infinitel man solutions. Graph the equations to provide more information The graphs are the same line, so the sstem has infinitel man solutions. + = - B + = Subtract the equations. + = - - ( + = ) The resulting equation is, so the sstem has solutions. Graph the equations to provide more information The graph shows that the lines are Houghton Mifflin Harcourt Publishing Compan and. Module Lesson 3

29 Your Turn Solve each sstem of linear equations b adding or subtracting = 3 - = = = -7 Eplain 3 Solving Linear Sstem Models b Adding or Subtracting Eample 3 Solve b adding or subtracting. A Perfect Patios is building a rectangular deck for a customer. According to the customer s specifications, the perimeter should be 0 meters and the difference between twice the length and twice the width should be meters. The sstem of equations l + w = 0 can be used to l - w = represent this situation, where l is the length and w is the width. What will be the length and width of the deck? Houghton Mifflin Harcourt Publishing Compan Image Credits: Huntstock, Inc/Alam Add the equations. l + w = 0 l - w = l + 0 = l = l = 11 Write the solution as an ordered pair. (l, w) = (11, 9) Substitute the value of l into one of the equations and solve for w. l + w = 0 (11) + w = 0 + w = 0 w = 1 w = 9 The length of the deck will be 11 meters and the width will be 9 meters. Module Lesson 3

30 B A video game and movie rental kiosk charges $ for each video game rented, and $1 for each movie rented. One da last week, a total of 11 video games and movies were rented for a total of $177. The sstem of equations + = 11 represents this situation, where + = 177 represents the number of video games rented and represents the number of movies rented. Find the numbers of video games and movies that were rented. Subtract the equations. + = 11 - ( + = 177) Substitute the value of into one of the equations and solve for. Write the solution as an ordered pair. video games and movies were rented. Your Turn 9. The perimeter of a rectangular picture frame is 6 inches. The difference of the length of the frame and twice its width is 1. The sstem of equations + w = 6 l represents this situation, l - w = 1 where l represents the length in inches and w represents the width in inches. What are the length and the width of the frame? Elaborate 10. How can ou decide whether to add or subtract to eliminate a variable in a linear sstem? Eplain our reasoning. 11. Discussion When a linear sstem has no solution, what happens when ou tr to solve the sstem b adding or subtracting? Houghton Mifflin Harcourt Publishing Compan 1. Essential Question Check-In When ou solve a sstem of linear equations b adding or subtracting, what needs to be true about the variable terms in the equations? Module Lesson 3

31 Evaluate: Homework and Practice 1. Which method of elimination would be best to solve the sstem of linear equations? Eplain. 1_ + 3 _ = _ = 1 Online Homework Hints and Help Etra Practice Solve each sstem of linear equations b adding or subtracting = = = = -. + = 5-3 = = - - = = = = = Houghton Mifflin Harcourt Publishing Compan. 6-3 = 15-3 = = = 0 Module Lesson 3

32 10. 1_ - 7_ 9 = - _ = _ + 7_ 9 = 6 _ = = = = = = = b = c a a - b = c 16. The sum of two numbers is 65, and the difference of the numbers is 7. The sstem of linear equations + = 65 represents this situation, where is the larger number and is the smaller - = 7 number. Solve the sstem to find the two numbers. Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

33 17. A rectangular garden has a perimeter of 10 feet. The length of the garden is feet greater than twice the width. The sstem of linear equations + w = 10 l l - w = represents this situation, where l is the length of the garden and w is its width. Find the length and width of the garden. 1. The sum of two angles is 90. The difference of twice the larger angle and the smaller angle is 105. The sstem of linear equations + = 90 represents this - = 105 situation where is the larger angle and is the smaller angle. Find the measures of the two angles. Houghton Mifflin Harcourt Publishing Compan 19. Ma and Sasha eercise a total of 0 hours each week. Ma eercises 15 hours less than times the number of hours Sasha eercises. The sstem of equations + = 0 represents this - = -15 situation, where represents the number of hours Ma eercises and represents the number of hours Sasha eercises. How man hours do Ma and Sasha eercise per week? 0. The sum of the digits in a two-digit number is 1. The digit in the tens place is more than the digit in the ones place. The sstem of linear equations + = 1 represents this situation, - = where is the digit in the tens place and is the digit in the ones place. Solve the sstem to find the two-digit number. Module Lesson 3

34 1. A pool compan is installing a rectangular pool for a new house. The perimeter of the pool must be 9 feet, and the length must be feet more than twice the width. The sstem of linear equations + w = 9 l represents l = w + this situation, where l is the length and w is the width. What are the dimensions of the pool?. Use one solution, no solutions, or infinitel man solutions to complete each statement. a. When the solution of a sstem of linear equations ields the equation =, the sstem has. b. When the solution of a sstem of linear equations ields the equation =, the sstem has. c. When the solution of a sstem of linear equations ields the equation 0 =, the sstem has. H.O.T. Focus on Higher Order Thinking 3. Multiple Representations You can use subtraction to solve the sstem of linear equations shown. + = - - = -10 Instead of subtracting - = -10 from + = -, what equation can ou add to get the same result? Eplain. Houghton Mifflin Harcourt Publishing Compan Image Credits: Marlene Ford/Alam Module Lesson 3

35 . Eplain the Error Liang s solution of a sstem of linear equations is shown. Eplain Liang s error and give the correct solution. 3 - = = = = -0 = - = = 1 3 (-) - = = 1 - = = -1 Solution: (-, -1) 5. Represent Real-World Problems For a school pla, Rico bought 3 adult tickets and 5 child tickets for a total of $0. Sasha bought 1 adult ticket and 5 child tickets for a total of $5. The sstem of linear equations + 5 = = 5 represents this Houghton Mifflin Harcourt Publishing Compan Image Credits: aerogondo/shutterstock situation, where is the cost of an adult ticket and is the cost of a child ticket. How much will Julia pa for 5 adult tickets and 3 child tickets? Module Lesson 3

36 Lesson Performance Task A local charit run has a Youth Race for runners under the age of 1. The entr fee is $5 for an individual or $ each for two runners from the same famil. Carter is collecting the registration forms and fees. After everone has registered, he picks up the cash bo and finds a dollar on the ground. He checks the cash bo and finds that it contains $00 and the registration slips for 7 runners. Does the dollar belong in the cash bo or not? Eplain our reasoning. (Hint: You can use the sstem of equations i + f = 7 and 5i + f = 00, where i equals the number of individual tickets and f equals the number of famil tickets.) Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

37 Name Class Date 11. Solving Linear Sstems b Multipling First Essential Question: How can ou solve a sstem of linear equations b using multiplication and elimination? Resource Locker Eplore 1 Understanding Linear Sstems and Multiplication A sstem of linear equations in which one of the like terms in each equation has either the same or opposite coefficients can that be readil solved b elimination. How do ou solve the sstem if neither of the pairs of like terms in the equations have the same or opposite coefficients? A B C Graph and label the following sstem of equations. - = 1 + = The solution to the sstem is. When both sides of an equation are multiplied b the same value, the equation [is/is not] still true D Multipl both sides of the second equation b. Houghton Mifflin Harcourt Publishing Compan E F G Write the resulting sstem of equations. - = 1 Graph and label the new sstem of equations. Solution: Can the new sstem of equations be solved using elimination now that appears in each equation? Module Lesson

38 Reflect 1. Discussion How are the graphs of + = and + = related?. Discussion How are the equations + = and + = related? Eplore Proving the Elimination Method with Multiplication The previous eample illustrated that rewriting a sstem of equations b multipling a constant term b one of the equations does not change the solutions for the sstem. What happens if a new sstem of equations is written b adding this new equation to the untouched equation from the original sstem? A Original Sstem New Sstem - = 1 + = - = 1 + = Add the equations in the new sstem. - = 1 + = B Write a new sstem of equations using this new equation. - = 1 C Graph and label the equations from this new sstem of equations. D Is the solution to this new sstem of equations the same as the solution to the original sstem of equations? Eplain E If the original sstem is A + B = C and D + E = F, where A, B, C, D, E, and F are constants, then multipl the second equation b a nonzero constant k to get kd + ke = kf. Add this new equation to A + B = C. A + B = C + kd + ke = kf + = F So, the original sstem is + B = C A, and the new sstem is D + E = F A + B = C. Houghton Mifflin Harcourt Publishing Compan Module Lesson

39 G Let ( 1, 1 ) be the solution to the original sstem. Fill in the missing parts of the following proof to show that ( 1, 1 ) is also the solution to the new sstem. H A 1 + B 1 = Given. I D 1 + E 1 = Given. J (D 1 + E 1 ) = kf Propert of Equalit K kd 1 + ke 1 = kf L + kd 1 + ke 1 = C + kf Propert of Equalit M A kd 1 + ke 1 = C + kf Substitute A 1 + for on the left. N A B 1 + ke 1 = C + kf Propert of Addition O (A 1 + kd 1 ) + ( B 1 + ke 1 ) = C + kf Propert of Addition P (A + kd) 1 + ( ) 1 = C + kf Q Therefore, ( 1, 1 ) is the solution to the new sstem. Reflect 3. Discussion Is a proof required using subtraction? What about division? Houghton Mifflin Harcourt Publishing Compan Eplain 1 Solving Linear Sstems b Multipling First In some sstems of linear equations, neither variable can be eliminated b adding or subtracting the equations directl. In these sstems, ou need to multipl one or both equations b a constant so that adding or subtracting the equations will eliminate one or more of the variables. Steps for Solving a Sstem of Equations b Multipling First 1. Decide which variable to eliminate.. Multipl one or both equations b a constant so that adding or subtracting the equations will eliminate the variable. 3. Solve the sstem using the elimination method. Module Lesson

40 Eample 1 Solve each sstem of equations b multipling. Check the answers b graphing the sstems of equations. 3 + = 7 A - = -10 Multipl the second equation b. ( - = -10) - = -0 Add the result to the first equation. 3 + = = = -33 Solve for. 11 = -33 = -3 Substitute -3 for in one of the original equations, and solve for. 3 + = 7 3 (-3) + = = 7 = 16 = The solution to the sstem is (-3, ) = B - 13 = 5 Multipl the first equation b and multipl the second equation b so the terms in the sstem have coefficients of -1 and 1 respectivel. (-3 + = ) -1 + = ( - 13 = 5 ) 1 - = Add the resulting equations = +1 - = = _ Houghton Mifflin Harcourt Publishing Compan Solve for. = = Module Lesson

41 Solve the first equation for when = = -3 + ( ) = -3 + = -3 = = The solution to the sstem is. - Your Turn Solve each sstem of equations b multipling. Check the answers b graphing the sstems of equations.. + = = Houghton Mifflin Harcourt Publishing Compan = = Module Lesson

42 Eplain Solving Linear Sstem Models b Multipling First You can solve a linear sstem of equations that models a real-world eample b multipling first. Eample Solve each problem b multipling first. A B Jessica spent $16.30 to bu 16 flowers. The bouquet contained daisies, which cost $1.75 each, and tulips, which cost $0.5 each. The sstem of equations d + t = 16 models this situation, where 1.75d + 0.5t = d is the number of daisies and t is the number of tulips. How man of each tpe of flower did Jessica bu? Multipl the first equation b -0.5 to eliminate t from each equation. Then, add the equations (d + t) = -0.5(16) -0.5d - 0.5t = d + 0.5t = d + 0.5t = d =.70 Find t. d + t = t = 16 t = 13 d = 3 The solution is (3, 13). Jessica bought 3 daisies and 13 tulips. The Tran famil is bringing 15 packages of cheese to a group picnic. Cheese slices cost $.50 per package. Cheese cubes cost $1.75 per package. The Tran famil spent a total of $30 on cheese. The sstem of equations s + c = 15 represents this situation, where s is the number of.50s c = 30 packages of cheese slices and c is the number of packages of cheese cubes. How man packages of each tpe of cheese did the Tran famil bu? Multipl the first equation b a constant so that c can be eliminated from both equations, and then subtract the equations. (s + c) = (15) s + c =.50s c = 30 -(.50s c ) = -30 Find c. s + c = 15 Your Turn + c = 15 s = s = c = The solution is. The Tran famil bought packages of sliced cheese and packages of cheese cubes. 6. Jacob s famil bought adult tickets and student tickets to the school pla for $6. Tatianna s famil bought 3 adult tickets and 3 students tickets for $60. The sstem of equations a + s = 6 3a + 3s = 60 models this situation, where a is the cost of an adult ticket and s is the cost of a student ticket. How much does each tpe of ticket cost? Houghton Mifflin Harcourt Publishing Compan Image Credits: Corbis Module Lesson

43 Elaborate 7. When would ou solve a sstem of linear equations b multipling?. How can ou use multiplication to solve a sstem of linear equations if none of the coefficients are multiples or factors of an of the other coefficients? 9. Essential Question Check-In How do ou solve a sstem of equations b multipling? Evaluate: Homework and Practice For each linear equation, a. find the product of 3 and the linear equation; Online Homework Hints and Help Etra Practice b. solve both equations for = = 1 Houghton Mifflin Harcourt Publishing Compan = 1. - = 13 Module Lesson

44 For each linear sstem, multipl the first equation b and add the new equation to the second equation. Then, graph this new equation along with both of the original equations = -1 + = = = Solve each sstem of linear equations b multipling. Verif each answer b graphing the sstem of equations = = = = Houghton Mifflin Harcourt Publishing Compan - Module 11 5 Lesson

45 = 13 - = = = Solve each sstem of linear equations using multiplication = 5-3 = 1 Houghton Mifflin Harcourt Publishing Compan = = 1 Solve each problem b multipling first. 13. The sum of two angles is 10. The difference between twice the larger angle and three times the smaller angle is 150. The sstem of equations + = 10-3 = 150 models this situation, where is the measure of the larger angle and is the measure of the smaller angle. What is the measure of each angle? Module Lesson

46 1. The perimeter of a rectangular swimming pool is 16 feet. The difference between the length and the width is 39 feet. The sstem of equations + = 16 - = 39 models this situation, where is the length of the pool and is the width of the pool. Find the dimensions of the swimming pool. 15. Jamian bought a total of 0 bagels and donuts for a morning meeting. He paid a total of $ Each donut cost $0.65 and each bagel cost $1.15. The sstem of equations b + d = 0 models this situation, where b is the number of bagels and 1.15b d = d is the number of donuts. How man of each did Jamian bu? 16. A clothing store is having a sale on shirts and jeans. shirts and pairs of jeans cost $6. 3 shirts and 3 pairs of jeans cost $7. The sstem of equations s + j = 6 models this situation, where s is the cost of a shirt and j is the cost of 3s + 3j = 7 a pair of jeans. How much does one shirt and one pair of jeans cost? 17. Jace bought 5 bath towels and returned hand towels. His sister Jana bought 3 bath towels and returned hand towels. Jace paid a total of $1 and Jana paid a total of $. The sstem of equations - h = 1 5b models this situation, where b is the 3b - h = price of a bath towel and h is the price of a hand towel. How much does each kind of towel cost? Houghton Mifflin Harcourt Publishing Compan Module 11 5 Lesson

47 1. Apples cost $0.95 per pound and bananas cost $1.10 per pound. Leah bought a total of pounds of apples and bananas for $.05. The sstem of equations a + b = models this 0.95a b =.05 situation, where a is the number of pounds of apples and b is the number of pounds of bananas. How man pounds of each did Leah bu? 19. Which of the following are possible was to eliminate a variable b multipling first? - + = 3-5 = -3 Houghton Mifflin Harcourt Publishing Compan Image Credits: Barbara Lechner/Shutterstock a. Multipl the first equation b. b. Multipl the first equation b 5 and the second equation b. c. Multipl the first equation b and d. Multipl the first equation b 5 and the second equation b. the second equation b. e. Multipl the first equation b and f. Multipl the second equation b. the second equation b Eplain the Error A linear sstem has two equations A + B = C and D + E = F. A student begins to solve the equation as shown. What is the error? A + B = C + k (D + E) = F (A + kd) + ( B + ke ) = C + F 1. Critical Thinking Suppose ou want to eliminate in this sstem: + 11 = = B what numbers would ou need to multipl the two equations in order to eliminate? Wh might ou choose to eliminate instead? H.O.T. Focus on Higher Order Thinking. Justif Reasoning Solve the following sstem of equations b multipling. + 3 = -1 + = -3 Would it be easier to solve the sstem b using substitution? Eplain our reasoning. Module Lesson

48 3. Multi-Step The school store is running a promotion on school supplies. Different supplies are placed on two shelves. You can purchase 3 items from shelf A and from shelf B for $16. Or ou can purchase items from shelf A and 3 from shelf B for $1. This can be represented b the following sstem of equations. a. Solve the sstem of equations 3A + B = 16 b multipling first. A + 3B = 1 b. If the supplies on shelf A are normall $6 each and the supplies on shelf B are normall $3 each, how much will ou save on each package plan from part A? Lesson Performance Task A chemist has a bottle of 1% acid solution and a bottle of 5% acid solution. She wants to mi the two solutions to get 100 ml of a % acid solution. a. Complete the table to write the sstem of equations. 1% Solution + 5% Solution = % Solution Amount of Solution (ml) Amount of Acid (ml) + = = 0.0(100) b. Solve the sstem of equations to find how much she will use from each bottle to get 100 ml of a % acid solution. Houghton Mifflin Harcourt Publishing Compan Image Credits: wacpan/ Shutterstock Module Lesson