Overview of Section.5 Introduction Sstems of linear equations were solved in Algebra using substitution, elimination, and graphing. These same techniques are applied to nonlinear sstems in this lesson. The first part of the lesson involves solving a sstem consisting of a quadratic equation and a linear equation. The different intersection possibilities are discussed. Eample involves a circle centered at the origin and a line. If students have not taken Geometr, the will not be familiar with the equation of a circle. You ma choose to do an additional eample of graphing a circle, for instance, + = 9. The last eample involves sstems of two quadratics. Graphing technolog is used to solve the sstems. Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations Resources Graphing calculators can be used for man of the problems in this lesson. Students should be familiar with how to resize the viewing window and/or zoom to an area of the graph. Use the intersection function on the graphing calculator to determine the point(s) of intersection. Formative Assessment Tips Opposing Views: This technique allows students to compare two or more solutions or was of thinking about a problem, or as is the case in this lesson, what variable should be solved for in a sstem involving a quadratic and a linear equation. Do ou solve for a -term and substitute, or do ou solve for an -term and substitute? Students should recognize that in substituting for the -term the are left with a single quadratic equation in one variable,. To be informative, students must present more than es/no answers. Students must be able to eplain their reasoning for classmates to hear. Once the different views are presented, all students should decide which view most closel reflects their own and give justification for their thinking. Students can respond in writing or ou can solicit viewpoints as a discussion prompt. MP Construct Viable Arguments and Critique the Reasoning of Others: Using this formative assessment technique allows ou to check students conceptual knowledge and their abilit to construct a viable argument. Pacing Suggestion Important discussion and understanding can occur during the eplorations. Take time to listen to students eplanations and then begin the formal lesson. Section.5 T-0
Common Core State Standards HSA-CED.A. Represent constraints b equations and b sstems of equations and interpret solutions as viable or nonviable options in a modeling contet. HSA-REI.C.7 Solve a simple sstem consisting of a linear equation and a quadratic equation in two variables algebraicall and graphicall. HSA-REI.D. Eplain wh the -coordinates of the points where the graphs of the equations = f() and = g() intersect are the solutions of the equation f() = g(); find the solutions approimatel, e.g., using technolog to graph the functions, make tables of values,. Include cases where f() and/or g() are linear, polnomial, functions. Eploration Motivate The signal from a radio station has a range of 50 miles. Another radio station is located 80 miles north and 00 miles east of the first radio station. This second station transmits in a range of 00 miles. Have students graph two circles that represent the ranges of the two stations. Ask students what will happen when the are driving in their car and reach the overlapping region of the two circles. The equations of the two circles represent a quadratic sstem. In this lesson, students will investigate some characteristics of the solutions of quadratic sstems. Discuss What does it mean to solve a sstem of equations? Answer will var. Listen for an understanding that solutions of a sstem are the ordered pairs which are solutions of all of the equations in the sstem. In Algebra, students worked with linear sstems. The ma describe solutions as the point where two lines intersect. What methods have ou used to solve sstems of equations? graphing, substitution, and elimination Could a sstem of equations include equations that are not linear? es Eploration In this eploration, students should be able to easil determine the three sstems that involve a parabola and a line. Using their knowledge of what information is known from a linear equation in slope-intercept form or a quadratic equation in standard form, students should be able to match the sstem with the graph. MP Construct Viable Arguments and Critique the Reasoning of Others: How did ou match the sstem with the graph? Listen for well-communicated justifications for students answers. How man solutions are there for a sstem with a quadratic function and a linear function? There could be two, one, or none. How man solutions are there for a sstem with two quadratic functions? There could be two, one, or none. Students ma believe that two quadratic functions could have a maimum of four solutions, as shown. The equation of the parabola with the horizontal line of smmetr is not a function. Eploration Think-Pair-Share: Give partners time to think about how the would solve the sstem without graphing. 0 0 The sstem can be solved using the table feature on a graphing calculator. The sstem is also easil solved b substitution. Communicate Your Answer Check that students understand the difference between the three approaches. If time permits, have students solve the sstem in Question. Connecting to Net Step Discussion of the reasoning used b students to do the matching in Eploration will help ou assess students recall of graphs of linear and quadratic equations as well as their understanding of a sstem of equations. Transition to the formal lesson when students finish the eplorations. T- Chapter
MAKING SENSE OF PROBLEMS.5 To be proficient in math, ou need to plan a solution pathwa rather than simpl jumping into a solution attempt. Solving Nonlinear Sstems Essential Question How can ou solve a nonlinear sstem of equations? Solving Nonlinear Sstems of Equations Work with a partner. Match each sstem with its graph. Eplain our reasoning. Then solve each sstem using the graph. a. = b. = + c. = 5 = + = + = + d. = + e. = + f. = + + = + = + = + + A. C. E. B. D. 5 F. 9 Solving Nonlinear Sstems of Equations Work with a partner. Look back at the nonlinear sstem in Eploration (f). Suppose ou want a more accurate wa to solve the sstem than using a graphical approach. a. Show how ou could use a numerical approach b creating a table. For instance, ou might use a spreadsheet to solve the sstem. b. Show how ou could use an analtical approach. For instance, ou might tr solving the sstem b substitution or elimination. Communicate Your Answer. How can ou solve a nonlinear sstem of equations?. Would ou prefer to use a graphical, numerical, or analtical approach to solve the given nonlinear sstem of equations? Eplain our reasoning. = + = + Section.5 Solving Nonlinear Sstems 8 8 5 7 9 Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Lesson Planning Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations. a. A; The graph shows a linear function with a positive slope and -intercept, and a parabola with the verte at the origin; (, ), (, ) b. E; The graph shows a linear function with a positive slope and -intercept, and a parabola with the verte at = ; (, 0), (, ) c. F; The graph shows a linear function with a negative slope and a positive -intercept, and a parabola with verte at = ; (, ), (, ) d. B; The graph shows two parabolas, each with verte at = ; (, 0), (, 0) e. C; The graph shows two parabolas, each with verte at = ; (, 0) f. D; The graph shows two parabolas, one with verte at =, and the other with verte at = ; (, 0), (, 9 ). a. See Additional Answers. b. + + = + + ; + = 0; ( )( + ) = 0; = 0 or + = 0; = or =. graphing, elimination, or substitution. Sample answer: Analtical because solutions that are not integers ma be difficult or impossible to find using other methods. Section.5
English Language Learners Vocabular Conve that the prefi non- means not. Emphasize that a nonlinear equation represents a graph that is not a line. Caution students from assuming that all nonlinear relationships are nonlinear functions because sometimes a nonlinear relationship cannot be characterized as a function. Etra Eample Solve the sstem b graphing. = 8 + =.5 Lesson What You Will Learn Core Vocabular sstem of nonlinear equations, p. Previous sstem of linear equations circle Solve sstems of nonlinear equations. Solve quadratic equations b graphing. Sstems of Nonlinear Equations Previousl, ou solved sstems of linear equations b graphing, substitution, and elimination. You can also use these methods to solve a sstem of nonlinear equations. In a sstem of nonlinear equations, at least one of the equations is nonlinear. For instance, the nonlinear sstem shown has a quadratic equation and a linear equation. = + = + 5 Equation is nonlinear. Equation is linear. When the graphs of the equations in a sstem are a line and a parabola, the graphs can intersect in zero, one, or two points. So, the sstem can have zero, one, or two solutions, as shown. No solution One solution Two solutions When the graphs of the equations in a sstem are a parabola that opens up and a parabola that opens down, the graphs can intersect in zero, one, or two points. So, the sstem can have zero, one, or two solutions, as shown. (, 0) No solution One solution Two solutions (, 0) Solving a Nonlinear Sstem b Graphing Solve the sstem b graphing. = Equation = Equation SOLUTION Graph each equation. Then estimate the point of intersection. The parabola and the line appear to intersect at the point (0, ). Check the point b substituting the coordinates into each of the original equations. Equation Equation = = =? (0) (0) =? (0) = = The solution is (0, ). (0, ) Chapter Quadratic Equations and Comple Numbers Teacher Actions Define a sstem of nonlinear equations. Note that the sstem could have more than two equations. Turn and Talk: What are the intersection possibilities for a parabola and a line? two, one, or no intersections two parabolas? two, one, or no intersections MP5 Use Appropriate Tools Strategicall: When ou use a graphing calculator to graph in the standard viewing window, it is not obvious what the solution is. Students should be proficient with changing the viewing window, adjusting for a better view. Chapter
Solving a Nonlinear Sstem b Substitution Solve the sstem b substitution. + = Equation + = Equation SOLUTION Begin b solving for in Equation. = + Solve for in Equation. Net, substitute + for in Equation and solve for. + = Write Equation. + ( + ) = Substitute + for. + = Simplif. Differentiated Instruction Auditor Review the solution strategies for solving sstems of linear equations. Eplain that these same solution strategies can be used to solve sstems of nonlinear equations. Pair students and ask them to work with their partners to discuss the steps for each solution method. Check 8 Intersection X=- Y=7 + = 0 Write in standard form. ( + )( ) = 0 Factor. + = 0 or = 0 Zero-Product Propert = or = Solve for. To solve for, substitute = and = into the equation = +. = + = () + = 7 Substitute for. = + = + = Substitute for. Etra Eample Solve the sstem b substitution. + = 5 + = 7 (, 9) and (, ) Check The solutions are (, 7) and (, ). Check the solutions b graphing the sstem. Solving a Nonlinear Sstem b Elimination Solve the sstem b elimination. 5 = Equation SOLUTION + + = 0 Equation Add the equations to eliminate the -term and obtain a quadratic equation in. 5 = + + = 0 = Add the equations. + = 0 Write in standard form. = ± 5 Use the Quadratic Formula. Because the discriminant is negative, the equation + = 0 has no real solution. So, the original sstem has no real solution. You can check this b graphing the sstem and seeing that the graphs do not appear to intersect. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem using an method. Eplain our choice of method.. = +. + + = 0. + = = + 8 + = 5 + = Etra Eample Solve the sstem b elimination. + = 0 + = no real solution MONITORING PROGRESS. (, 0); Sample answer: substitution because the equations were in general form and one could be substituted for the other. no solution: Sample answer: elimination because the equations were arranged with like terms in the same column. (, 9 ) and (0, ); Sample answer: elimination because the equations were arranged with like terms in the same column Section.5 Solving Nonlinear Sstems Teacher Actions Opposing Views: Ask the Probing Question, In Eample, there is an -term. Must ou solve for an -term and substitute, or can ou solve for a -term and substitute? Eplain. Give students time to consider their answer. Some students will sa es, some will sa no, and some will be uncertain. Ask a few students from each viewpoint to share their thinking. The class should decide that substituting for a -term will leave a quadratic equation in one variable,. Similar to Eample, eliminating the -terms in Eample leaves a quadratic equation in one variable,. Section.5
Etra Eample Solve the sstem b substitution. + = 0 = + (, ) and (, ) MONITORING PROGRESS. (0, ) and (, 0) 5. no solution Center Radius: r Point on circle: (, ) Some nonlinear sstems have equations of the form + = r. This equation is the standard form of a circle with center (0, 0) and radius r. When the graphs of the equations in a sstem are a line and a circle, the graphs can intersect in zero, one, or two points. So, the sstem can have zero, one, or two solutions, as shown. No solution One solution Two solutions. ( 5, 5 ) and (, 0) Solving a Nonlinear Sstem b Substitution Solve the sstem b substitution. + = 0 Equation = + 0 Equation SOLUTION COMMON ERROR You can also substitute = in Equation to find. This ields two apparent solutions, (, ) and (, ). However, (, ) is not a solution because it does not satisf Equation. You can also see (, ) is not a solution from the graph. Substitute + 0 for in Equation and solve for. + = 0 Write Equation. + ( + 0) = 0 Substitute + 0 for. + 9 0 + 00 = 0 0 0 + 90 = 0 Epand the power. Write in standard form. + 9 = 0 Divide each side b 0. ( ) = 0 Perfect Square Trinomial Pattern = To find the -coordinate of the solution, substitute = in Equation. = () + 0 = Zero-Product Propert Check The solution is (, ). Check the solution b graphing the sstem. You can see that the line and the circle intersect onl at the point (, ). (, ) Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem.. + = 5. + =. + = = + = + = + Chapter Quadratic Equations and Comple Numbers Teacher Actions Before doing Eample, it will be necessar to review, or perhaps introduce, the equation of a circle with radius r. MP Reason Abstractl and Quantitativel: Wh is substitution a good solution strateg for the sstem in Eample? Answers will var. Listen for recognition that there are no like variable terms to eliminate. Graphing b hand is a possibilit when the solution involves integer coordinates. Turn and Talk: Do ou think ou could substitute the solution, =, into the nonlinear equation? One of the solutions to the nonlinear equation is not a solution to the linear equation. Chapter
ANOTHER WAY In Eample 5(a), ou can also find the solutions b writing the given equation as + = 0 and solving this equation using the Quadratic Formula. Solving Equations b Graphing You can solve an equation b rewriting it as a sstem of equations and then solving the sstem b graphing. Core Concept Solving Equations b Graphing Step To solve the equation f () = g(), write a sstem of two equations, = f () and = g(). Step Graph the sstem of equations = f () and = g(). The -value of each solution of the sstem is a solution of the equation f () = g(). Solving Quadratic Equations b Graphing Solve (a) + 5 = + + and (b) (.5) +.5 = ( +.5) b graphing. SOLUTION a. Step Write a sstem of equations using each side of the original equation. Equation Sstem = + 5 + 5 = + + = + + Step Use a graphing calculator to graph the sstem. Then use the intersect feature to find the -value of each solution of the sstem. Etra Eample 5 a. Solve the equation + 7 = + b graphing..59 and 0.9 b. Solve the equation ( ) + = ( ) b 9 graphing. = MONITORING PROGRESS 7. = 8. = and = Intersection X=-.759 Y=-.7 5 Intersection X=.5905 Y=.98 5 5 Intersection X=0 Y=0 The graphs intersect when.8 and 0.. The solutions of the equation are.8 and 0.. b. Step Write a sstem of equations using each side of the original equation. Equation (.5) +.5 = ( +.5) Sstem = (.5) +.5 = ( +.5) Step Use a graphing calculator to graph the sstem, as shown at the left. Then use the intersect feature to find the -value of each solution of the sstem. The graphs intersect when = 0. The solution of the equation is = 0. Monitoring Progress Solve the equation b graphing. Help in English and Spanish at BigIdeasMath.com 7. + 5 = ( ) + 8. ( + )( ) = + + Section.5 Solving Nonlinear Sstems 5 Teacher Actions Big Idea: The Core Concept is a Big Idea, and it is a technique that students will use often in mathematics! Discuss the technique. Thumbs Up: Could the technique be used to solve + 7 =? Hopefull all thumbs will be in the up position! MP Model with Mathematics: Graphing technolog makes it possible to solve a sstem of two quadratic equations. Using a good viewing window is helpful. Think-Pair-Share: Have students work independentl on Questions 7 and 8 and then share with partners. Discuss as a class. Closure I Used to Think But Now I Know: Take time for students to reflect on their current understanding of solving a nonlinear sstem. Section.5 5
Assignment Guide and Homework Check ASSIGNMENT Basic:,, odd, 7 5 odd, 7 odd, 5, 58, Average:, even, 5, 0 50 even, 5, 58, Advanced:, even, 5, 0 even, HOMEWORK CHECK Basic: 5, 5, 7, 9, 5 Average:, 8,, 8, Advanced: 0, 8,, 0,. There could be no solution, one solution, or two solutions.. = ; = + ; It is the onl linear sstem, and it is the onl sstem that has onl one solution.. (0, ) and (, 0). (, 5) 5. no solution. (, 8) and (, ) 7. (, ) 8. no solution 9. (, ) and (, ) 0. (, 0) and (0, ). (, ). no solution. (, ) and (9, ). (, 5) and (, 0) 5. (, 8) and (, ). (7, 0) and (0, 7) 7. (0, 8) 8. (, ) 9. no solution 0. (0, ) and (, ). (, ) and (, ). (, ) and (, 0). no solution. (, ) and (, ) 5. A and C. A; The sstem has no solution because the graphs do not intersect. Also, when using substitution or elimination, the result is a quadratic equation with imaginar solutions..5 Eercises Vocabular and Core Concept Check Chapter Quadratic Equations and Comple Numbers Dnamic Solutions available at BigIdeasMath.com. WRITING Describe the possible solutions of a sstem consisting of two quadratic equations.. WHICH ONE DOESN T BELONG? Which sstem does not belong with the other three? Eplain our reasoning. = + = + = = + In Eercises 0, solve the sstem b graphing. Check our solution(s). (See Eample.). = +. = ( ) + 5 = 0.5( + ) = 5 5. = +. = 0 7 = 5 = 7 7. = + 8 + 8 8. = 9 = 0 = 9. = ( ) 0. = ( + ) = + = + In Eercises, solve the sstem of nonlinear equations using the graph... 8.. 8 5 = + + = 5 + Monitoring Progress and Modeling with Mathematics + = = + In Eercises 5, solve the sstem b substitution. (See Eamples and.) 5. = + 5. + = 9 = + = 7 7. + = 8. = = 8 + = 8 9. + = 0. = + 5 + = =. =. + = 7 = + + = 0. + = 7. + = 5 + = + = 5. USING EQUATIONS Which ordered pairs are solutions of the nonlinear sstem? = 5 + = + A (, ) B (, 0) C (8,.5) D (7, 0). USING EQUATIONS How man solutions does the sstem have? Eplain our reasoning. = 7 + 9 = 7 + 5 A 0 B C D Chapter
hsnb_alg_pe_005.indd 7 In Eercises 7, solve the sstem b elimination. (See Eample.) 7. = 5 8. + 5 = + = 5 9. + = 8 + 9 = 8 5 0. = 0 = + 8 + + =. + =. = + + 7 = = + 7. = 0 7 = + 0 +. 0 + = 80 + 55 5 + = 0 85 5. ERROR ANALYSIS Describe and correct the error in using elimination to solve a sstem. = + = 0 = 8 = 8 = 7. NUMBER SENSE The table shows the inputs and outputs of two quadratic equations. Identif the solution(s) of the sstem. Eplain our reasoning. 9 9 9 7 5 7 9 9 57 9 In Eercises 7, solve the sstem using an method. Eplain our choice of method. 7. = 8. = = + + + = 7 9. + 0 + = 0. = 0.5 0 = 0 = +. = ( ) + ( ) + = 0. + = 00 = + USING TOOLS In Eercises 8, solve the equation b graphing. (See Eample 5.). + = +. = + 0 5. ( + )( ) = + 7. 5 = + 8 + 95 7. ( ) = ( + )( + 9) 8 8. ( + )( + 8) = ( + )( + ) 9. REASONING A nonlinear sstem contains the equations of a constant function and a quadratic function. The sstem has one solution. Describe the relationship between the graphs. 50. PROBLEM SOLVING The range (in miles) of a broadcast signal from a radio tower is bounded b a circle given b the equation + = 0. A straight highwa can be modeled b the equation = + 0. For what lengths of the highwa are cars able to receive the broadcast signal? 5. PROBLEM SOLVING A car passes a parked police car and continues at a constant speed r. The police car begins accelerating at a constant rate when it is passed. The diagram indicates the distance d (in miles) the police car travels as a function of time t (in minutes) after being passed. Write and solve a sstem of equations to find how long it takes the police car to catch up to the other car. t = 0 t =? r = 0.8 mi/min d =.5t Section.5 Solving Nonlinear Sstems 7 /5/5 0:50 AM Dnamic Teaching Tools Dnamic Assessment & Progress Monitoring Tool Interactive Whiteboard Lesson Librar Dnamic Classroom with Dnamic Investigations 7. (, 7) and (0, 5) 8. no solution 9. no solution 0. (, ) and (, ). about (.5,.7) and about (0.5, 5.9). about (.59,.) and about ( 5.,.). (, ) and (, ). (, 5) 5. The terms that were added were not like terms; 0 = + 0; = 7 or = 0. (, 9) and (7, 9); For these -values, each equation has the same -value. 7. (0, ); Sample answer: elimination because the equations are arranged with like terms in the same column 8. no solution; Sample answer: substitution because the first equation can be substituted into the second equation 9. about (., 0) and about (5., 0); Sample answer: substitution because the second equation can be substituted into the first equation 0. (, ) and (, ); Sample answer: elimination because the equations are arranged with like terms in the same column. (, ) and (5, ); Sample answer: graphing because substitution and elimination would require more steps in this case. about (., 0.57); Sample answer: substitution because the second equation can be substituted into the first equation. = 0.. and.5 5. 0. and.7. = and = 5 7. = and = 8. no solution 9. The graphs intersect at the verte of the quadratic function. 50. from ( 8, ) to (, 8), a length of 0 5.9 miles 5. d = 0.8t; d =.5t ; 0. min Section.5 7
5. Sample answer: = ; = + ; 5. THOUGHT PROVOKING Write a nonlinear sstem that has two different solutions with the same -coordinate. Sketch a graph of our sstem. Then solve the sstem. 58. HOW DO YOU SEE IT? The graph of a nonlinear sstem is shown. Estimate the solution(s). Then describe the transformation of the graph of the linear function that results in a sstem with no solution. = 5. OPEN-ENDED Find three values for m so the sstem has no solution, one solution, and two solutions. Justif our answer using a graph. = + 8 7 = m + (, 0) (, 0) = + (, 0) and (, 0) 5. See Additional Answers. 5. MAKING AN ARGUMENT You and a friend solve the sstem shown and determine that = and =. You use Equation to obtain the solutions (, ), (, ), (, ), and (, ). Your friend uses Equation to obtain the solutions (, ) and (, ). Who is correct? Eplain our reasoning. + = 8 Equation = 0 Equation 55. COMPARING METHODS Describe two different was ou could solve the quadratic equation. Which wa do ou prefer? Eplain our reasoning. + 7 = + 59. MODELING WITH MATHEMATICS To be eligible for a parking pass on a college campus, a student must live at least mile from the campus center. mi Main Street (0, 0) mi mi campus center Oak Lane 5 mi College Drive Mini-Assessment. Solve the sstem b graphing. = + 5 = + 7 8 (, 8) (, 5) 5. ANALYZING RELATIONSHIPS Suppose the graph of a line that passes through the origin intersects the graph of a circle with its center at the origin. When ou know one of the points of intersection, eplain how ou can find the other point of intersection without performing an calculations. 57. WRITING Describe the possible solutions of a sstem that contains (a) one quadratic equation and one equation of a circle, and (b) two equations of circles. Sketch graphs to justif our answers. Maintaining Mathematical Proficienc a. Write equations that represent the circle and Oak Lane. b. Solve the sstem that consists of the equations in part (a). c. For what length of Oak Lane are students not eligible for a parking pass? 0. CRITICAL THINKING Solve the sstem of three equations shown. + = = + = + Solve the inequalit. Graph the solution on a number line. (Skills Review Handbook). > 8. + 7. ( ) Write an inequalit that represents the graph. (Skills Review Handbook). 5.. Reviewing what ou learned in previous grades and lessons (, 8) and (, 5). Solve the sstem b substitution. + + = 0 = (, 0) and (, ) 8 Chapter Quadratic Equations and Comple Numbers. Solve the sstem b elimination. + = 5 + = 7 no real solution. Solve the sstem b substitution. + = 7 = + (, ) and (, ) 5. Solve the equation b graphing. a. + = 0.5 + 7 0. and.8 b. (.5) +.5 = ( +.5).7 If students need help... Resources b Chapter Practice A and Practice B Puzzle Time Student Journal Practice Differentiating the Lesson Skills Review Handbook If students got it... Resources b Chapter Enrichment and Etension Cumulative Review Start the net Section 8 Chapter