SECTION 11.6 Systems of Nonlinear Equations 821 11.6 Systems of Nonlinear Equations Now Work the ‘Are You Prepared?’ problems on page 827. • Lines ( Section 1.5 , pp. 32 – 43 ) • Circles ( Section 1.6 , pp. 48 – 52 ) • Parabolas (Section 10.2, pp. 682–687) • Ellipses (Section 10.3, pp. 692–699) • Hyperbolas (Section 10.4, pp. 705–714) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Solve a System of Nonlinear Equations Using Substitution (p. 821) 2 Solve a System of Nonlinear Equations Using Elimination (p. 822) In Section 11.1, we observed that the solution to a system of linear equations could be found geometrically by determining the point(s) of intersection (if any) of the equations in the system. Similarly, in solving systems of nonlinear equations, the solution(s) also represent(s) the point(s) of intersection (if any) of the graphs of the equations. There is no general method for solving a system of nonlinear equations. Sometimes substitution is best; other times elimination is best; and sometimes neither of these methods works. Experience and a certain degree of imagination are your allies here. Before we begin, two comments are in order. • If the system contains two variables and if the equations in the system are easy to graph, then graph them. By graphing each equation in the system, you can get an idea of how many solutions a system has and approximate their location. • Extraneous solutions can creep in when solving nonlinear systems, so it is imperative to check all apparent solutions. 1 Solve a System of Nonlinear Equations Using Substitution Algebraic Solution Using Substitution Graphing Solution Use a graphing utility to graph each equation. If necessary for your graphing utility, solve each equation for y: = + Y x3 2 1 and = Y x2 . 2 2 Figure 20 shows the graphs of the equations using a TI-84 Plus CE. Observe that the system apparently has two solutions. Use INTERSECT to find that the solutions to the system of equations are ( ) −0.5,0.5 and ( ) 2,8 . To use substitution to solve the system, we choose to solve equation (1) for y. − = − = + x y y x 3 2 3 2 Equation (1) First, notice that the system contains two variables and that we know how to graph each equation by hand. See Figure 19. The system apparently has two solutions. Solving a System of Nonlinear Equations Solve the following system of equations: − =− − = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ (1) (2) x y x y 3 2 2 0 2 (continued) EXAMPLE 1 A line Aparabola Figure 20 10 22 26 6 Y2 5 2x2 Y 1 5 3x 1 2 Figure 19 x y –6 6 3x – y = –2 (y = 3x + 2) 2x2 – y = 0 (y = 2x2) 10 –2 (2, 8) ( ) ,1 – 2 1 – 2 –
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