11.1 Solving Linear Systems by Graphing

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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