Essential Question How can you solve a nonlinear system of equations?
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1 .5 Solving Nonlinear Sstems Essential Question 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. B. 8 C. D. 8 E. 5 F MAKING SENSE OF PROBLEMS To be proficient in math, ou need to plan a solution pathwa rather than simpl jumping into a solution attempt. 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. 7 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
2 .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. No solution One solution Two solutions Solving a Nonlinear Sstem b Graphing Solve the sstem b graphing. = Equation = Equation 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) =? (0) = = The solution is (0, ). Chapter Quadratic Equations and Comple Numbers
3 Solving a Nonlinear Sstem b Substitution Solve the sstem b substitution. + = Equation + = Equation Check Intersection X=- Y=7 8 Begin b solving for in Equation. = + Solve for in Equation. Net, substitute + for in Equation and solve for. + = Write Equation. + ( + ) = Substitute + for. + = Simplif. + = 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. 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 + + = 0 Equation Check 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. + = = = 5 + = Section.5 Solving Nonlinear Sstems
4 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. Center Radius: r Point on circle: (, ) 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 Solving a Nonlinear Sstem b Substitution Solve the sstem b substitution. + = 0 Equation = + 0 Equation 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 = = 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
5 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(). 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 Quadratic Equations b Graphing Solve (a) + 5 = + + and (b) (.5) +.5 = ( +.5) b graphing. a. Step Write a sstem of equations using each side of the original equation. Equation Sstem = = + + = + + Step Use a graphing calculator to graph the sstem. Then use the intersect feature to find the -value of each solution of the sstem. Intersection X=-.759 Y=-.7 5 Intersection X=.5905 Y= 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 = ( ) + 8. ( + )( ) = + + Section.5 Solving Nonlinear Sstems 5
6 .5 Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. 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. = + = + = = + = + + = = = + Monitoring Progress and Modeling with Mathematics In Eercises 0, solve the sstem b graphing. Check our solution(s). (See Eample.). = +. = ( ) + 5 = 0.5( + ) = 5 5. = +. = 0 7 = 5 = 7 7. = = 9 = 0 = In Eercises 5, solve the sstem b substitution. (See Eamples and.) 5. = = 9 = + = = 8. = = 8 + = = 0. = = = 9. = ( ) 0. = ( + ) = + = + In Eercises, solve the sstem of nonlinear equations using the graph =. + = 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. = = A 0 B C D Chapter Quadratic Equations and Comple Numbers
7 In Eercises 7, solve the sstem b elimination. (See Eample.) 7. = = + = = = = 0 = =. + =. = = = + 7. = 0 7 = = = 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 In Eercises 7, solve the sstem using an method. Eplain our choice of method. 7. = 8. = = = 7. = ( ) + ( ) + = 0. + = 00 = + USING TOOLS In Eercises 8, solve the equation b graphing. (See Eample 5.). + = +. = ( + )( ) = = ( ) = ( + )( + 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 = 0. = = 0 = + Section.5 Solving Nonlinear Sstems 7
8 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. = = m + 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 = 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 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). > ( ) Write an inequalit that represents the graph. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons Chapter Quadratic Equations and Comple Numbers
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