SECTION 11.6 Systems of Nonlinear Equations 825 Solving a System of Nonlinear Equations Solve: − = − + = ⎧ ⎨ ⎪ ⎩⎪⎪ xy y x y 3 2 2 9 4 10 2 2 2 ( ) ( ) 1 2 Algebraic Solution Multiply equation (1) by 2, and add the result to equation (2), to eliminate the y2 terms. − =− + = ⎧ ⎨ ⎪⎪⎪ ⎩ ⎪⎪⎪ + = + = xy y x y x xy x xy 6 4 4 9 4 10 9 6 6 3 2 2 2 2 2 2 2 Since ≠ x 0 (do you see why?), solve for y in this equation to get = − ≠ y x x x 2 3 2 0 2 (3) Now substitute for y in equation (2) of the system. + = x y 9 4 10 2 2 Equation (2) ( ) + − = x x x 9 4 2 3 2 10 2 2 2 Substitute = − y x x 2 3 2 . 2 + − + = x x x x 9 4 12 9 10 2 2 4 2 Expand and simplify. + − + = x x x x 9 4 12 9 10 4 2 4 2 Multiply both sides by x . 2 − + = x x 18 22 4 0 4 2 Subtract x 10 2 from both sides. − + = x x 9 11 2 0 4 2 Divide both sides by 2. This quadratic equation (in x2) can be factored: ( )( ) − − = − = − = = = = ± = ± = ± x x x x x x x x 9 2 1 0 9 2 0 or 1 0 2 9 1 2 9 2 3 1 2 2 2 2 2 2 To find y, use equation (3). • If = = − = − ⋅ = = x y x x 2 3 : 2 3 2 2 2 3 2 2 3 4 2 2 2 2 • =− = − = − − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = − =− x y x x If 2 3 : 2 3 2 2 2 3 2 2 3 4 2 2 2 2 • If = = − = − ⋅ = − x y x x 1: 2 3 2 2 3 1 2 1 2 2 2 • If ( ) = − = − = − − − = x y x x 1: 2 3 2 2 3 1 2 1 2 2 2 The system has four solutions: ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ − − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 2 3 , 2 , 2 3 , 2 , ( ) − 1, 1 2 , and ( ) −1, 1 2 . Check them for yourself. Graphing Solution Graph each equation using a graphing utility. If necessary for your graphing utility, solve each equation for y. To solve equation (1) for y, we view the equation as a quadratic equation in the variable y. − = − − − = xy y y xy 3 2 2 2 3 2 0 2 2 Place in standard form. ( ) ( ) ( ) = − − ± − − ⋅ ⋅ − ⋅ y x x 3 3 4 2 2 2 2 2 Use the quadratic formula with = =− =− a b x c 2, 3 , 2. = ± + y x x 3 9 16 4 2 Simplify. Using a graphing utility, graph = + + Y x x 3 9 16 4 1 2 and = − + Y x x 3 9 16 4 . 2 2 From equation (2), graph = − Y x 10 9 2 3 2 and = − − Y x 10 9 2 . 4 2 Figure 26(a) shows the graphs using a TI-84 Plus CE. Figure 26(b) EXAMPLE 5 (1) (2) Add. Divide each side by 3. Use INTERSECT to find that the solutions to the system of equations are ( ) ( ) −1, 0.5 , 0.47, 1.41 , ( ) − 1, 0.5, and ( ) − − 0.47, 1.41 , each rounded to two decimal places. Figure 26(b) shows the solutions obtained using Desmos. Figure 26(a) 2.5 22.5 24 4 Y4 5 10 2 9x2 2 2 Y3 5 10 2 9x2 2 Y2 5 9x2 1 16 3x 2 4 Y1 5 9x2 1 16 3x 1 4 Now Work PROBLEM 47
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