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LESSON9.1?Solving Linear Systemsby GraphingA-REI.3.6Solve systems of linearequations exactly andapproximately (e.g., withgraphs), focusing on pairsof linear equations in twovariables. Also A-CED.1.3ESSENTIAL QUESTIONHow can you find the solution of a system of linear equations by graphing?Solving a Linear System by GraphingA system of linear equations, also called a linear system, consists of two or morelinear equations that have the same variables. A solution of a system of linearequations with two variables is an ordered pair that satisfies all of the equations inthe system. The values of the variables in the ordered pair make each equation inthe system true.Math On the Spotmy.hrw.comSystems of linear equations can be solved by graphing and by using algebraicmethods. In this lesson you will learn to solve linear systems by graphing theequations in the system and analyzing how those graphs are related.EXAMPLE 1A-REI.3.6Solve the system of linear equations below by graphing. Check your answer.-x y 32x y 6{ Houghton Mifflin Harcourt Publishing CompanySTEP 1STEP 2Find the intercepts for each equation, plus a third point for a check.Graph each line.-x y 32x y 6x-intercept: 3x-intercept: 3y-intercept: 3y-intercept: 6third point: (3, 6)third point: (-1, 8)yFind the point of intersection.8The two lines appear to intersectat (1, 4).6-x y 34STEP 3Check to see if (1, 4) makes bothequations true.-x y 32x y 6?3-(1) 4 ?62(1) 4 3 3 2-2O2x y 624x6-26 6 The solution is (1, 4).Lesson 9.1277
yYOUR TURNPersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com1. Solve the system of linear equationsbelow by graphing. Check your answer.x y 8x - y 2{8642Solution:xO2468-2Special Systems of Linear EquationsWhen a system of equations consists of two lines that intersect in one point, asshown in Figure 1 below, there is exactly one solution to the system.Math On the Spotmy.hrw.comIf two linear equations in a system have the same graph, as shown in Figure 2, thegraphs are coincident lines, or the same line. There are infinitely many solutions ofthe system because every point on the line represents a solution of both equations.A system with at least one solution is a consistent system. Consistent systems caneither be independent or dependent. An independent system has exactly one solution. The graph of anindependent system consists of two intersecting lines. A dependent system has infinitely many solutions. The graph of a dependentsystem consists of two coincident lines.The table below summarizes how systems of linear equations can be classified.Classification of Systems of Linear EquationsClassificationConsistent andIndependentConsistent andDependentInconsistentNumber ofSolutionsExactly oneInfinitely manyNoneDescriptionDifferent slopesSame slope,same y-interceptSame slope,differenty-interceptsGraphFigure 1278Unit 2AFigure 2Figure 3 Houghton Mifflin Harcourt Publishing CompanyWhen the two lines in a system do not intersect, as shown in Figure 3, they areparallel lines. There are no ordered pairs that satisfy both equations, so there is nosolution. A system that has no solution is an inconsistent system.
EXAMPLE 2A-REI.3.6Use the graph to solve each system of linear equations. Classify each system.A{x y 72x 2y 68The lines do not intersect and appearto be parallel lines.{3x - y 16x y 342x 2y 6This system has no solution. Thesystem is inconsistent.BMyMyNotesNotesyx y 72x2x 2y 6x y 3-4-2The equations have the same graph,so the graphs are coincident lines.O246-2This system has infinitely many solutions. The system is consistent anddependent.Math TalkREFLECT2. Use the graph above to identify two lines that represent a linear systemwith exactly one solution. What are the equations of the lines? Explainyour reasoning.Mathematical PracticesWhy is the greatest possiblenumber of solutions onefor a system of linearequations whose graphshows two distinctlines?YOUR TURN Houghton Mifflin Harcourt Publishing CompanyGraph each system of linear equations. Classify each system.3.{ 2xx - 2yy 484.y{ yy 2x2x - 46y64422x-2O-2-424x6-4-2O24-2-4PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 9.1279
Estimating a Solution by GraphingYou can estimate a solution for a linear system by graphing and then check yourestimate to determine if it is an approximate solution.Math On the Spotmy.hrw.comEXAMPLE 3A-REI.3.6Estimate the solution for the linear system by graphing.x 2y 22x - 3y 12STEP 1AnimatedMathmy.hrw.comSTEP 2{yGraph each equation byfinding intercepts.2-2Ox 2y 22x - 3y 12x-intercept: 2x-intercept: 6-2y-intercept: 1y-intercept: -4-42x4682x - 3y 12-6Find the point of intersection.-8The two lines appear tointersect at about 4 1, -1 1 .STEP 3x 2y 2( 4 8)Check if ( 4 14 , -1 18 ) is an approximate solution.x 2y 2? 24 41 2 (-1 81) ? 24 14 (-2 14) (2 2 )2x - 3y 12? 122(4 41) - 3 (-1 81) ? 128 1 - (-3 3 ) 2811 78 12 The point 4 14 , -1 18 does not make both equations true, butit is acceptable since 11 78 is close to 12. So, 4 14 , -1 18 is anapproximate solution.)yYOUR TURN5. Estimate the solution for the linearsystem by graphing.x-y 3x 2y 4{Approximate solution:PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com280Unit 2A42x-2O-2-4246 Houghton Mifflin Harcourt Publishing Company(
Guided PracticeSolve the system of linear equations by graphing. Check your answer. (Example 1)1.{ 2xx - yy 27y2x-y 72x y he solution of the system isx-2 O-2246-4.Does your solution check in both original equations?-6-8Use the graph to solve and classify each system of linear equations. (Example 2)2.-1{-2xx 2yy -23.Is the system consistent?-x y 3{ -2x 2y -2y-x y 34Is the system consistent?2Is the system independent?-2x 2y -2x-4 -2 O-2Solution:-424x y -12x 2y -2Solution:Estimate the solution for the system of linear equations by graphing. (Example 3) Houghton Mifflin Harcourt Publishing Company4.{x y -12x - y 5y2xx y -12x - y 5x-intercept:x-intercept:-2 O-2y-intercept:y-intercept:-446-6The two lines appear to intersect at.?2-8ESSENTIAL QUESTION CHECK-IN5. How does graphing help you solve a system of linear equations?Lesson 9.1281
NameClassDate9.1 Independent PracticePersonalMath TrainerA-REI.3.6, A-CED.1.3my.hrw.comSolve each system of linear equations bygraphing. Check your answer.x - y -26. 2x y 8Estimate the solution for the linear system bygraphing.x y 59. x - 3y 3{y866442-2 O-22x24682{ 2xx -4yy -5-4yy82x-8 -6 -4 -2 O-26242-2 O-2Solution:{ -2xx - 4y2y 4-811.y2-8 -6 -4 -2 O-2x2x2468{ 4x2x -- 2yy -6-12y8-44-8-8 -4 O-4-6-8Unit 2A8{42826Graph each system. Then classify it as consistentand independent, consistent and dependent, orinconsistent.x 2y 610.x 26Solution:4Approximate solution:88.x-2 O-2Solution:7.y8x48 Houghton Mifflin Harcourt Publishing Company{OnlineAssessment andIntervention
12. Use the graphed system below to find anapproximate solution. Then write a system ofequations that represents the system.y15. Suppose you use the graph of a system oflinear equations to estimate the solution.Explain how you would check your estimateto determine if it is an approximate solution.8642-2 O-2x246813. Communicate Mathematical Ideas Whena system of linear equations is graphed, howis the graph of each equation related to thesolutions of that equation?16. Without graphing, describe the graph of thislinear system of equations.{ xy 34What is the solution of the linear system? Houghton Mifflin Harcourt Publishing Company14. Critical Thinking Write sometimes, always,or never to complete the following statement:If the equations in a system of linearequations have the same slope, there areinfinitely many solutions for thesystem.17. How can you recognize a dependent systemof equations by analyzing the equations in thesystem?18. How would you classify a system of equationswhose graph is composed of two lines withdifferent slopes and the same y-intercepts?What is the solution to the system?Lesson 9.1283
Amount Saved19. Sophie and Marcos are each saving for new bicycles. So far, Sophie has 10saved and can earn 5 per hour walking dogs. Marcos has 4 saved andcan earn 8 per hour at his family’s plant nursery. After how many hoursof work will Sophie and Marcos have saved theysame amount? What will that amount be? Use40the system of equations below to complete the30graph.Sophie: y 5x 10Marcos: y 8x 42010xO246Hours Worked20. Represent Real-World Problems Cora ran 1 mile last week and will run7 miles per week from now on. Hana ran 2 miles last week and will run 4 milesy 7x 1per week from now on. The system of linear equations y 4x 2 can beused to represent this situation. Explain what x and y represent in the equations.{21. Explain the Error Jake was asked to give an example of an inconsistent systemof linear equations. He wrote the system shown below. Explain Jake’s error.y 2x - 4y x-4{22. Draw Conclusions The equations in a system of linear equations havedifferent slopes and the same y-intercept. Can you find the solution withoutgraphing? Explain.284Unit 2AWork Area Houghton Mifflin Harcourt Publishing Company Image Credits: Art man/ShutterstockFOCUS ON HIGHER ORDER THINKING
Math Trainer Online Assessment and Intervention Personal my.hrw.com y x O 2 4 6 8 8 6 2 4-2 Math On the Spot my.hrw.com 1. Solve the system of linear equations below by graphing. Check your answer. { x y 8 x-y 2 Solution: YOUR TURN Special Systems of Linear Equations When a system of
