9.6 Solving Nonlinear Systems Of Equations
9.6Solving Nonlinear Systemsof EquationsEssential QuestionHow can you solve a system of two equationswhen one is linear and the other is quadratic?Solving a System of EquationsWork with a partner. Solve the systemof equations by graphing each equationand finding the points of intersection.y64System of Equationsy x 2Lineary x2 2xQuadratic2 6 4 2246x 2Analyzing Systems of EquationsWork with a partner. Match each system of equations with its graph. Then solve thesystem of equations.a. y x2 4y x 2b. y x2 2x 2y 2x 2c. y x2 1y x 1d. y x2 x 6y 2x 2A.B.10 101010 10 10C. 10D.10 1010To be proficient in math,you need to analyzegivens, relationships,and goals.10 1010 10 10MAKING SENSEOF PROBLEMS10Communicate Your Answer3. How can you solve a system of two equations when one is linear and the otheris quadratic?4. Write a system of equations (one linear and one quadratic) that has (a) nosolutions, (b) one solution, and (c) two solutions. Your systems should bedifferent from those in Explorations 1 and 2.Section 9.6hsnb alg1 pe 0906.indd 525Solving Nonlinear Systems of Equations5252/5/15 9:01 AM
9.6 LessonWhat You Will LearnSolve systems of nonlinear equations by graphing.Solve systems of nonlinear equations algebraically.Core VocabulVocabularylarrysystem of nonlinear equations,p. 526Previoussystem of linear equationsApproximate solutions of nonlinear systems and equations.Solving Nonlinear Systems by GraphingThe methods for solving systems of linear equations can also be used to solve systemsof nonlinear equations. A system of nonlinear equations is a system in which at leastone of the equations is nonlinear.When a nonlinear system consists of a linear equation and a quadratic equation, thegraphs can intersect in zero, one, or two points. So, the system can have zero, one, ortwo solutions, as shown.No solutionsOne solutionTwo solutionsSolving a Nonlinear System by GraphingSolve the system by graphing.y 2x2 5x 1Equation 1y x 3Equation 2SOLUTIONy 2x 2 5x 1Step 1 Graph each equation.Step 2 Estimate the point of intersection.The graphs appear to intersectat ( 1, 4). 6Step 3 Check the point from Step 2 bysubstituting the coordinates intoeach of the original equations.Equation 1y 26y x 3( 1, 4) 6Equation 2 5x 1? 4 2( 1)2 5( 1) 1 4 1 3 4 4 4 42x2 y x 3? The solution is ( 1, 4).Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSolve the system by graphing.1. y x2 4x 4y 2x 5526Chapter 9hsnb alg1 pe 0906.indd 5262. y x 6y 2x2 x 33. y 3x 15y —12 x2 2x 7Solving Quadratic Equations2/5/15 9:01 AM
Solving Nonlinear Systems AlgebraicallyREMEMBERSolving a Nonlinear System by SubstitutionThe algebraic proceduresthat you use to solvenonlinear systems aresimilar to the proceduresthat you used to solvelinear systems in Sections5.2 and 5.3.Solve the system by substitution.y x2 x 1Equation 1y 2x 3Equation 2SOLUTIONStep 1 The equations are already solved for y.Step 2 Substitute 2x 3 for y in Equation 1 and solve for x.CheckUse a graphing calculator tocheck your answer. Notice thatthe graphs have two points ofintersection at ( 4, 11) and (1, 1).14 2x 3 x2 x 1Substitute 2x 3 for y in Equation 1.3 x2 3x 1Add 2x to each side.0 x2 3x 4Subtract 3 from each side.0 (x 4)(x 1)Factor the polynomial.x 4 0x 4orx 1 0orx 1Zero-Product PropertySolve for x.Step 3 Substitute 4 and 1 for x in Equation 2 and solve for y.y 2( 4) 3 1212 2Substitute for x in Equation 2. 11y 2(1) 3 1Simplify.So, the solutions are ( 4, 11) and (1, 1).Solving a Nonlinear System by EliminationSolve the system by elimination.y x2 3x 2Equation 1y 3x 8Equation 2SOLUTIONStep 1 Because the coefficients of the y-terms are the same, you do notSneed to multiply either equation by a constant.CheckUse a graphing calculator tocheck your answer. The graphsdo not intersect.Step 2 Subtract Equation 2 from Equation 1.S8y x2 3x 2Equation 1y Equation 2 3x 80 x2 88 6Step 3 Solve for x.S0 x2 6 8 6 x2Resulting equation from Step 2Subtract 6 from each side.The square of a real number cannot be negative. So, the system hasno real solutions.Section 9.6hsnb alg1 pe 0906.indd 527Subtract the equations.Solving Nonlinear Systems of Equations5272/5/15 9:01 AM
Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSolve the system by substitution.4. y x2 95. y 5xy 9y x2 3x 36. y 3x2 2x 1y 5 3xSolve the system by elimination.7. y x2 x8. y 9x2 8x 6y x 5y 5x 49. y 2x 5y 3x2 x 4Approximating SolutionsWhen you cannot find the exact solution(s) of a system of equations, you can analyzeoutput values to approximate the solution(s).Approximating Solutions of a Nonlinear SystemApproximate the solution(s) of the system to the nearest thousandth.y —12 x2 3Equation 1y 3xEquation 2SOLUTION8ySketch a graph of the system. You can see that the system has one solutionbetween x 1 and x 2.Substitute 3x for y in Equation 1 and rewrite the equation.63x —12 x2 341y x2 323x —12 x2 3 0Substitute 3x for y in Equation 1.Rewrite the equation.2 2y 3x2xBecause you do not know how to solve this equation algebraically, letf (x) 3x —12 x2 3. Then evaluate the function for x-values between 1 and 2.f (1.1) 0.26f (1.2) 0.02Because f (1.1) 0 and f (1.2) 0, the zero is between 1.1 and 1.2.f (1.2) is closer to 0 than f (1.1), so decrease your guess and evaluate f (1.19).REMEMBERf (1.19) 0.012The function values thatare closest to 0 correspondto x-values that bestapproximate the zerosof the function.Because f (1.19) 0 and f (1.2) 0, the zero isbetween 1.19 and 1.2. So, increase guess.f (1.191) 0.009Result is negative. Increase guess.f (1.192) 0.006Result is negative. Increase guess.f (1.193) 0.003Result is negative. Increase guess.f (1.194) 0.0002Result is negative. Increase guess.f (1.195) 0.003Result is positive.Because f (1.194) is closest to 0, x 1.194.Substitute x 1.194 into one of the original equations and solve for y.y —12 x2 3 —12 (1.194)2 3 3.713So, the solution of the system is about (1.194, 3.713).528Chapter 9hsnb alg1 pe 0906.indd 528Solving Quadratic Equations2/5/15 9:01 AM
REMEMBERWhen entering theequations, be sure to usean appropriate viewingwindow that shows allthe points of intersection.For this system, anappropriate viewingwindow is 4 x 4and 4 y 4.Recall from Section 5.5 that you can use systems of equations to solve equations withvariables on both sides.Approximating Solutions of an EquationSolve 2(4)x 3 0.5x2 2x.SOLUTIONYou do not know how to solve this equation algebraically. So, use each side of theequation to write the system y 2(4)x 3 and y 0.5x2 2x.Method 1Use a graphing calculator to graph the system. Then use the intersectfeature to find the coordinates of each point of intersection.44 44 4IntersectionX -1Y 2.5 44IntersectionX .46801468 Y -.8265105 4One point of intersectionis ( 1, 2.5).The other point of intersectionis about (0.47, 0.83).So, the solutions of the equation are x 1 and x 0.47.Method 2Use the table feature to create a table of values for the equations. Find thex-values for which the corresponding y-values are approximately equal.XSTUDY TIP-1.03-1.02-1.01You can use thedifferences between thecorresponding y-valuesto determine the bestapproximation ofa solution.-.99-.98-.97X 9.50X .47When x 1, the correspondingy-values are -.7988-.8142-.8296-.8448-.86-.875When x 0.47, the correspondingy-values are approximately 0.83.So, the solutions of the equation are x 1 and x 0.47.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comUse the method in Example 4 to approximate the solution(s) of the system to thenearest thousandth.10. y 4xy 11. y 4x2 1x2 x 3y 2(3)x 12. y x2 3x4y x2 x 10Solve the equation. Round your solution(s) to the nearest hundredth.13. 3x 1 x2 2x 51 x()14. 4x2 x 2 —2 5Section 9.6hsnb alg1 pe 0906.indd 529Solving Nonlinear Systems of Equations5292/5/15 9:01 AM
Exercises9.6Dynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. VOCABULARY Describe how to use substitution to solve a system of nonlinear equations.2. WRITING How is solving a system of nonlinear equations similar to solving a system oflinear equations? How is it different?Monitoring Progress and Modeling with MathematicsIn Exercises 3–6, match the system of equations with itsgraph. Then solve the system.3. y x2 2x 14. y x2 3x 2y x 1y x 35. y x 16. y x 3y x2 x 1A.1B. 2y x214. y 3x2 4x 515. y x 7y 6x 316. y x2 7y x2 2x 117. y 5 x2yy 2x 418. y 2x2 3x 4y 54x213. y x 5y x2 2x 5yIn Exercises 13–18, solve the system by substitution.(See Example 2.)y 4x 2In Exercises 19–26, solve the system by elimination.(See Example 3.)4x2 519. y x2 5x 720. y 3x2 x 2y 5x 9C.D.yy42 41x23. y 2x 125. y 2x 0In Exercises 7–12, solve the system by graphing.(See Example 1.)7. y 3x2 2x 18. y x2 2x 5y x 7y 2x 59. y 2x2 4x10. y —2 x2 3x 4y 211. y 1—3 x2 y 2x530Chapter 9hsnb alg1 pe 0906.indd 5301y x 22x 3y 2x 224. y x2 x 1y x22x 222. y 2x2 x 3y 4x 224 421. y x2 2x 2y x 412. y 4x2 5x 7y 3x 5y x 226. y 2x 7y x2 4x 6y 5x x2 227. ERROR ANALYSIS Describe and correct the error insolving the system of equations by graphing. y x2 3x 4y 2x 4The only solutionof the systemis (0, 4).y4224xSolving Quadratic Equations2/5/15 9:01 AM
28. ERROR ANALYSIS Describe and correct the error insolving for one of the variables in the system. Exercise 37 using substitution. Compare the exactsolutions to the approximated solutions.y 3x2 6x 4y 448. COMPARING METHODS Solve the system iny 3(4)2 6(4) 4y 28Exercise 38 using elimination. Compare the exactsolutions to the approximated solutions.Substitute.Simplify.49. MODELING WITH MATHEMATICS The attendancesy for two movies can be modeled by the followingequations, where x is the number of days since themovies opened.In Exercises 29–32, use the table to describe thelocations of the zeros of the quadratic function f.29.30.31.32.x 4 3 2 101f (x) 22442 2x 101234f (x)1151 1 11x 4 3 2 101f (x)3 1 131123x123456 25 9153 5f (x)47. COMPARING METHODS Solve the system iny x2 35x 100Movie Ay 5x 275Movie BWhen is the attendance for each movie the same?50. MODELING WITH MATHEMATICS You and a friendare driving boats on the same lake. Your path canbe modeled by the equation y x2 4x 1, andyour friend’s path can be modeled by the equationy 2x 8. Do your paths cross each other? If so,what are the coordinates of the point(s) where thepaths meet?In Exercises 33–38, use the method in Example 4 toapproximate the solution(s) of the system to the nearestthousandth. (See Example 4.)33. y x2 2x 334. y 2x 5y 3x51. MODELING WITH MATHEMATICS The arch of ay x2 3x 135. y 2(4)x 1bridge can be modeled by y 0.002x2 1.06x,where x is the distance (in meters) from the leftpylons and y is the height (in meters) of the archabove the water. The road can be modeled by theequation y 52. To the nearest meter, how far fromthe left pylons are the two points where the roadintersects the arch of the bridge?36. y x2 4x 4y 5x 2y 3x2 8x37. y x2 x 538. y 2x2 x 8y 2x2 6x 3y x2 5In Exercises 39–46, solve the equation. Round yoursolution(s) to the nearest hundredth. (See Example 5.)39. 3x 1 x2 7x 140. x2y 0.002x 2 1.06xy 52 2x 2x 510042. 2x2 8x 10 x2 2x 51 x—2() x2 53 x()45. 8x 2 3 2 —244.1.5(2)x200300400500x52. MAKING AN ARGUMENT Your friend says that a 3 x2 4x46. 0.5(4)x 5x 6Section 9.6hsnb alg1 pe 0906.indd 53120010041. x2 6x 4 x2 2x43. 4ysystem of equations consisting of a linear equationand a quadratic equation can have zero, one, two,or infinitely many solutions. Is your friend correct?Explain.Solving Nonlinear Systems of Equations5312/5/15 9:01 AM
58. PROBLEM SOLVING The population of a country isCOMPARING METHODS In Exercises 53 and 54, solve the2 million people and increases by 3% each year. Thecountry’s food supply is sufficient to feed 3 millionpeople and increases at a constant rate that feeds0.25 million additional people each year.system of equations by (a) graphing, (b) substitution,and (c) elimination. Which method do you prefer?Explain your reasoning.53. y 4x 354. y x2 5y x2 4x 1a. When will the country first experience a foodshortage?y x 755. MODELING WITH MATHEMATICS The functionb. The country doubles the rate at which its foodsupply increases. Will food shortages still occur?If so, in what year?y x2 65x 256 models the number y ofsubscribers to a website, where x is the number ofdays since the website launched. The number ofsubscribers to a competitor’s website can be modeledby a linear function. The websites have the samenumber of subscribers on Days 1 and 34.59. ANALYZING GRAPHS Use the graphs of the linear andquadratic functions.ya. Write a linear function that models the numberof subscribers to the competitor’s website.Bb. Solve the system to verify the function frompart (a).A(2, 6)x56. HOW DO YOU SEE IT? The diagram shows thegraphs of two equations in a system that hasone solution.y x 2 6x 4ya. Find the coordinates of point A.b. Find the coordinates of point B.60. THOUGHT PROVOKING Is it possible for a systemof two quadratic equations to have exactly threesolutions? exactly four solutions? Explain yourreasoning. (Hint: Rotations of the graphs of quadraticequations still represent quadratic equations.)y cxa. How many solutions will the system have whenyou change the linear equation to y c 2?61. PROBLEM SOLVING Solve the system of threeequations shown.b. How many solutions will the system have whenyou change the linear equation to y c 2?y 2x 8y x2 4x 3y 3(2)x57. WRITING A system of equations consists of a62. PROBLEM SOLVING Find the point(s) of intersection,quadratic equation whose graph opens up and aquadratic equation whose graph opens down. Describethe possible numbers of solutions of the system.Sketch examples to justify your answer.if any, of the line y x 1 and the circlex2 y2 41.Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsGraph the system of linear inequalities. (Section 5.7)63. y 2xy x 464. y 4x 1y 765. y 3 2xy 5 3x66. x y 62y 3x 4Graph the function. Describe the domain and range. (Section 8.3)67. y 3x2 2532Chapter 9hsnb alg1 pe 0906.indd 53268. y x2 6x69. y 2x2 12x 770. y 5x2 10x 3Solving Quadratic Equations2/5/15 9:02 AM
Section 9.6 Solving Nonlinear Systems of Equations 527 Solving Nonlinear Systems Algebraically Solving a Nonlinear System by Substitution Solve the system by substitution. y x2 Equation 1 x 1 y 2x 3 Equation 2 SOLUTION Step 1 The equations are already solved for y. Step 2 Substitute 2x 3 for y in Equation 1 and solve
9.1 Properties of Radicals 9.2 Solving Quadratic Equations by Graphing 9.3 Solving Quadratic Equations Using Square Roots 9.4 Solving Quadratic Equations by Completing the Square 9.5 Solving Quadratic Equations Using the Quadratic Formula 9.6 Solving Nonlinear Systems of Equations 9 Solving Quadratic Equations
Outline Nonlinear Control ProblemsSpecify the Desired Behavior Some Issues in Nonlinear ControlAvailable Methods for Nonlinear Control I For linear systems I When is stabilized by FB, the origin of closed loop system is g.a.s I For nonlinear systems I When is stabilized via linearization the origin of closed loop system isa.s I If RoA is unknown, FB provideslocal stabilization
ods for solving nonlinear systems of equations that are com-binations of the nonlinear ABS methods and quasi-Newton methods. Another interesting class of methods have been proposed by Kublanovskaya and Simonova [8] for estimat-ing the roots of m nonlinear coupled algebraic equations
Nonlinear Finite Element Analysis Procedures Nam-Ho Kim Goals What is a nonlinear problem? How is a nonlinear problem different from a linear one? What types of nonlinearity exist? How to understand stresses and strains How to formulate nonlinear problems How to solve nonlinear problems
Third-order nonlinear effectThird-order nonlinear effect In media possessing centrosymmetry, the second-order nonlinear term is absent since the polarization must reverse exactly when the electric field is reversed. The dominant nonlinearity is then of third order, 3 PE 303 εχ The third-order nonlinear material is called a Kerr medium. P 3 E
Graphing Systems of Equations Review Partner Practice Friday January 11, 2019 Solving Systems of Equations by Substitution (Day 1) p. 6 - 7 Homework 2: Solving Systems by Substitution (Day 1) Monday January 14, 2019 Solving Systems of Equations by Substitution (Day 2) p. 8 - 9 Homework 3: Solving Systems by Substitution (Day 2) Tuesday January .
Introduction to Nonlinear Optics 1 1.2. Descriptions of Nonlinear Optical Processes 4 1.3. Formal Definition of the Nonlinear Susceptibility 17 1.4. Nonlinear Susceptibility of a Classical Anharmonic . Rabi Oscillations and Dressed Atomic States 301 6.6. Optical Wave Mixing in Two-Level Systems 313 Problems 326 References 327 7. Processes .
TOP SECRET//HCS/COMINT -GAMMA- /TK//FGI CAN GBR//RSEN/ORCON/REL TO USA, CAN, GBR//20290224/TK//FGI CAN GBR//RSEN/ORCON/REL TO USA, CAN, GBR//20290224 In the REL TO marking, always list USA first, followed by other countries in alphabetical trigraph order. Use ISO 3166 trigraph country codes; separate trigraphs with a comma and a space. The word “and” has been eliminated. DECLASSIFICATION .
