10.3 Solving Nonlinear Systems of Equations

Transcription

1 60 CHAPTER 0 Conic Sections Identif whether each equation, when graphed, will be a parabola, circle, ellipse, or hperbola. Then graph each equation =. = +. = =. 9-9 = = = 8. + = 6 9. = = = = =. = + 6. a + b + a - b = 0. Solving Nonlinear Sstems of Equations S Solve a Nonlinear Sstem b Substitution. Solve a Nonlinear Sstem b Elimination. In Section., we used graphing, substitution, and elimination methods to find solutions of sstems of linear equations in two variables. We now appl these same methods to nonlinear sstems of equations in two variables. A nonlinear sstem of equations is a sstem of equations at least one of which is not linear. Since we will be graphing the equations in each sstem, we are interested in real number solutions onl. Solving Nonlinear Sstems b Substitution First, nonlinear sstems are solved b the substitution method. EXAMPLE Solve the sstem e - = - = Solution We can solve this sstem b substitution if we solve one equation for one of the variables. Solving the first equation for is not the best choice since doing so introduces a radical. Also, solving for in the first equation introduces a fraction. We solve the second equation for. - = Second equation - = Solve for. Replace with - in the first equation, and then solve for. - = First equation $%& b - - = Replace with = - + = = 0 = or = Let = and then let = in the equation = - to find corresponding -values. Let =. Let =. = - = - = = - = - = 0 The solutions are (, ) and (, 0), or the solution set is,,, 06. Check both solutions in both equations. Both solutions satisf both equations, so both are solutions

2 Section 0. Solving Nonlinear Sstems of Equations 6 of the sstem. The graph of each equation in the sstem is shown net. Intersections of the graphs are at (, ) and (, 0). (, ) (, 0) Solve the sstem e - = + = -. EXAMPLE Solve the sstem = e + = 6 Solution This sstem is ideal for substitution since is epressed in terms of in the first equation. Notice that if =, then both and must be nonnegative if the are real numbers. Substitute for in the second equation, and solve for. + = 6 + = 6 + = = = 0 = - or = Let = The solution - is discarded because we have noted that must be nonnegative. To see this, let = - in the first equation. Then let = in the first equation to find a corresponding -value. Let = -. = = - Not a real number Let =. = = Since we are interested onl in real number solutions, the onl solution is,. Check to see that this solution satisfies both equations. The graph of each equation in the sstem is shown to the right. 6 (, ) 6 6 = - Solve the sstem e +. = 0

3 6 CHAPTER 0 Conic Sections EXAMPLE Solve the sstem e + = + = Solution We use the substitution method and solve the second equation for. + = Second equation = - Now we let = - in the first equation. + = $%& b - + = = = 0 First equation Let = -. B the quadratic formula, where a =, b = -6, and c =, we have = 6 { -6 - # # # = 6 { - Since - is not a real number, there is no real solution, or. Graphicall, the circle and the line do not intersect, as shown below. Solve the sstem e + = 9 - =. CONCEPT CHECK Without solving, how can ou tell that + = 9 and + = 6 do not have an points of intersection? Answer to Concept Check: + = 9 is a circle inside the circle + = 6, therefore the do not have an points of intersection. Solving Nonlinear Sstems b Elimination Some nonlinear sstems ma be solved b the elimination method. EXAMPLE Solve the sstem e + = 0 - =

4 Section 0. Solving Nonlinear Sstems of Equations 6 Solution We will use the elimination, or addition, method to solve this sstem. To eliminate when we add the two equations, multipl both sides of the second equation b -. Then + = 0 e - - = - # is equivalent to e + = = - = 9 = = { Add. Divide both sides b. To find the corresponding -values, we let = and = - in either original equation. We choose the second equation. Let =. - = - = - = = = { = { Let = -. - = - - = - = = = { = { The solutions are,, -,,, -, and -, -. Check all four ordered pairs in both equations of the sstem. The graph of each equation in this sstem is shown. (, ) (, ) (, ) 0 (, ) Solve the sstem e + = 6 - =. Vocabular, Readiness & Video Check Martin-Ga Interactive Videos Watch the section lecture video and answer the following questions.. In Eample, wh do we choose not to solve either equation for?. In Eample, what important reminder is made as the second equation is multiplied b a number to get opposite coefficients of? See Video 0.

5 6 CHAPTER 0 Conic Sections 0. Eercise Set MIXED Solve each nonlinear sstem of equations for real solutions. See Eamples through.. e + = + = 0. e + = + = 0. e + = 0 =. e + = 0 = = -. e - = 6. e + = + = - + = 9 7. e 6 - = 6 8. e + = + = 9. e + = - = 0. e + = - = 6 = -. e - = 6 = +. e - = =. e + = 0. e 6 - = = = +. e + = - 6. e + = 9 + = 7. e = - = - 8. e = - = - 9. e + = - + = 0. e - = - + = + =. e + + =. e + = - =. e = + = - +. e = - - = -. e + = 9 - = 9 6. e + = = - 7. e + = 6 - = 0 8. e + = = = 6 9. c = = 6 0. c = - +. e = + = REVIEW AND PREVIEW. e = + = 0 Graph each inequalit in two variables. See Section Find the perimeter of each geometric figure. See Section inches ( ) inches ( 0) inches 9. ( ) meters 0. feet feet ( 7) feet meters ( ) feet ( ) centimeters

6 Section 0. Nonlinear Inequalities and Sstems of Inequalities 6 CONCEPT EXTENSIONS For the eercises below, see the Concept Check in this section.. Without graphing, how can ou tell that the graph of + = and + = do not have an points of intersection?. Without solving, how can ou tell that the graphs of = + and = + 7 do not have an points of intersection?. How man real solutions are possible for a sstem of equations whose graphs are a circle and a parabola? Draw diagrams to illustrate each possibilit.. How man real solutions are possible for a sstem of equations whose graphs are an ellipse and a line? Draw diagrams to illustrate each possibilit. Solve.. The sum of the squares of two numbers is 0. The difference of the squares of the two numbers is. Find the two numbers. 6. The sum of the squares of two numbers is 0. Their product is 8. Find the two numbers. 7. During the development stage of a new rectangular kepad for a securit sstem, it was decided that the area of the rectangle should be 8 square centimeters and the perimeter should be 68 centimeters. Find the dimensions of the kepad. 8. A rectangular holding pen for cattle is to be designed so that its perimeter is 9 feet and its area is feet. Find the dimensions of the holding pen. Recall that in business, a demand function epresses the quantit of a commodit demanded as a function of the commodit s unit price. A suppl function epresses the quantit of a commodit supplied as a function of the commodit s unit price. When the quantit produced and supplied is equal to the quantit demanded, then we have what is called market equilibrium. Demand function Market equilibrium Suppl function 9. The demand function for a certain compact disc is given b the function p = and the corresponding suppl function is given b p = where p is in dollars and is in thousands of units. Find the equilibrium quantit and the corresponding price b solving the sstem consisting of the two given equations. 0. The demand function for a certain stle of picture frame is given b the function p = and the corresponding suppl function is given b p = 9 + where p is in dollars and is in thousands of units. Find the equilibrium quantit and the corresponding price b solving the sstem consisting of the two given equations. Use a graphing calculator to verif the results of each eercise.. Eercise.. Eercise.. Eercise.. Eercise. 0. Nonlinear Inequalities and Sstems of Inequalities S Graph a Nonlinear Inequalit. Graph a Sstem of Nonlinear Inequalities. Graphing Nonlinear Inequalities We can graph a nonlinear inequalit in two variables such as in a wa 6 similar to the wa we graphed a linear inequalit in two variables in Section.7. First, graph the related equation =. The graph of the equation is our 6 boundar. Then, using test points, we determine and shade the region whose points satisf the inequalit. EXAMPLE Graph 6. Solution of 6 First, graph the equation 6 (Continued on net page) includes the graph of 6 =. Sketch a solid curve since the graph =. The graph is an ellipse, and it