7.4 Systems of Nonlinear Equations in Two …

1 Section of Nonlinear Equations in Two Variables767 Objectives Recognize Systems ofnonlinear Equations in twovariables. Solve Nonlinear Systems bysubstitution. Solve Nonlinear Systems byaddition. Solve problems usingsystems of of Nonlinear Equations in Two VariablesScientists debate the prob-ability that a doomsdayrock will collide with has been estimated that anasteroid, a tiny planet thatrevolves around the sun,crashes into Earth about onceevery 250,000 years, and thatsuch a collision would havedisastrous results. In 1908, a small fragmentstruck Siberia, leveling thousands of acres of theory about the extinction of dinosaurs65 million years ago involves Earth s collisionwith a large asteroid and the resulting drasticchanges in Earth s the path of Earth and the pathof a comet is essential to detecting threatening space debris.

2 Orbits about the sunare not described by linear Equations in the form The ability to solvesystems that contain Nonlinear Equations provides NASA scientists watching fortroublesome asteroids with a way to locate possible collision points withEarth s of Nonlinear Equations and Their SolutionsA system oftwo Nonlinear equationsin two variables, also called a nonlinearsystem, contains at least one equation that cannot be expressed in the formHere are two examples:A solutionof a Nonlinear system in two variables is an ordered pair of realnumbers that satisfies both Equations in the system .

3 The solution setof the system isthe set of all such ordered pairs. As with linear Systems in two variables, the solutionof a Nonlinear system (if there is one) corresponds to the intersection point(s) of thegraphs of the Equations in the system . Unlike linear Systems , the graphs can becircles, parabolas, or anything other than two lines. We will solve Nonlinear systemsusing the substitution method and the addition +10x2+y2= +3 Not in the formAx + By = term x2 isnot equation is inthe form Ax + By = terms x2 and y2 arenot +By= +By= Recognize Systems of nonlinearequations in two ExercisesExercises 66 68 will help you prepare for the material covered inthe next by the substitution method:y= +3y= by the addition and in the samerectangular coordinate system .

4 What are the two intersectionpoints? Show that each of these ordered pairs satisfies +1y+322=4x-y=3 3x+5y=-3. 2x+4y=-4bbP-BLTZMC07_727-804-hr 21-11-2008 12:58 Page 767768 Chapter 7 Systems of Equations and inequalities Solve Nonlinear Systems a variable Using the Substitution MethodThe substitution method involves converting a Nonlinear system into one equationin one variable by an appropriate substitution. The steps in the solution process areexactly the same as those used to solve a linear system by substitution. However,when you obtain an equation in one variable , this equation may not be linear.

5 In ourfirst example, this equation is a Nonlinear system by the Substitution MethodSolve by the substitution method:bx2=2y+103x-y= 1(The graph is a parabola.)(The graph is a line.)SolutionStep 1 Solve one of the Equations for one variable in terms of the beginby isolating one of the variables raised to the first power in either of the solving for in the second equation, which has a coefficient of we can is the second equation in the given to both 9 from both 2 Substitute the expression from step 1 into the other substitutefor in the first gives us an equation in one variable .

6 NamelyThe variable has been 3 Solve the resulting equation containing one is the equation containing one the distributive numerical terms on the all terms to one side and set thequadratic equation equal to each factor equal to for Step 4 Back-substitute the obtained values into the equation from step we have the of the solutions, we back-substitute 4 for and 2 forinto the equation If x is 4, y=3142-9=3,so 14, 32 is a x is 2, y=3122-9=-3,so 12, -32 is a x=4 x=2 x-4=0 or x-2=0 1x-421x-22=0 x2-6x+8=0 x2=6x-8 x2=6x-18+10 x2=213x-92+10yx2=213x-92+ 3x-9x2=2 y +10y3x-9 3x-9=yy 3x=y+9 3x-y=9-1,yP-BLTZMC07_727-804-hr 21-11-2008 12:58 Page 768 Section of Nonlinear Equations in Two Variables769 Step 5 Check the proposed solutions in both of the system s given by checking (4, 3).

7 Replace with 4 and with are the given and statements ordered pair (4, 3) satisfies both Equations . Thus, (4, 3) is a solution of the let s check Replace with 2and with in both given are the given and statements ordered pair also satisfies both Equations and is a solution of the solutions are (4, 3) and and the solution set is Figure the graphs of the Equations in the system and the solutionsas intersection Point1 Solve by the substitution method:Solving a Nonlinear system by the Substitution MethodSolve by the substitution method.

8 Bx-y=31x-222+1y+322= 2bx2=y-14x-y= , 32, 12, , -32,12, -329=9, 4=4, 4 -6+10 6+3 9y= 22 21-32+10 3122-1-32 9 x2=2y+10 3x-y=9-3yx12, , 16=16, 16 6+10 12-3 9y= 42 2132+10 3142-3 9 x2=2y+10 3x-y=9yx 11234 2 3 4 9 8 7 6 512345 1 2 3 4 5yx3x y = 9x2 = 2y + 10(4, 3)(2, 3)Figure of intersectionillustrate the Nonlinear system ssolutions.(The graph is a line.)(The graph is a circle.)SolutionGraphically, we are finding the intersection of a line and a circle withcenter and radius 1 Solve one of the Equations for one variable in terms of the will solvefor in the linear equation that is, the first equation.

9 (We could also solve for )This is the first equation in the given to both 2 Substitute the expression from step 1 into the other for in the second gives an equation in one variable , namelyThe variable has been +3-222+1y+322= y+3( x-2)2+(y +3)2=4xy+3y x=y+3 x-y= , -32 Study TipRecall thatdescribes a circle with center and radius , k21x-h22+1y-k22=r2P-BLTZMC07_727-804-hr 21-11-2008 12:58 Page 769770 Chapter 7 Systems of Equations and InequalitiesStep 3 Solve the resulting equation containing one is the equation containing one numerical terms in the first the formula tosquare and Combine like terms on the 4 from both sides and set the quadraticequation equal to out each variable factor equal to for Step 4 Back-substitute the obtained values into the equation from step we have the of the solutions.

10 We back-substitute for and for in the equation Step 5 Check the proposed solutions in both of the system s given moment to show that each ordered pair satisfies both given Equations ,and . The solutions are and and thesolution set of the given system is Figure the graphs of the Equations in the system and the solutionsas intersection Point2 Solve by the substitution method:Eliminating a variable Using the Addition MethodIn solving linear Systems with two variables, we learned that the addition methodworks well when each equation is in the form For Nonlinear Systems ,the addition method can be used when each equation is in the formIf necessary, we will multiply either equation or both Equations byappropriate numbers so that the coefficients of or will have a sum of 0.