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Chapter 6 Conic Sections Section 6.4 Nonlinear Systems of Equations Nonlinear Systems of Equations • The graphs of the equations in a nonlinear system of equations can have no point of intersection or one or more points of intersection. • The coordinates of each point of intersection represent a solution of the system of equations. • When no point of intersection exists, the system of equations has no real-number solution. • We can solve nonlinear systems of equations by using the substitution or elimination method. Example • Solve the following system of equations: x y 9 2x y 3 2 2 Example continued • We use the substitution method. First, we solve equation (2) for y. Example continued • Next, we substitute y = 2x 3 in equation (1) and solve for x: Example continued • Now, we substitute these numbers for x in equation (2) and solve for y. • x=0 x = 12 / 5 Example continued Check: (0, 3) x2 y 2 9 2x y 3 0 3 9 99 2(0) (3) 3 33 2 3 12 9 , Check: 5 5 x2 y 2 9 2x y 3 12 2 5 9 2 5 9 99 2( 125 ) ( 95 ) 3 33 • Visualizing the Solution Example • Solve the following system of equations: xy = 4 3x + 2y = 10 Example continued Solve xy = 4 for y. Substitute into 3x + 2y = 10. Example continued • Use the quadratic formula to solve: 3 x 10 x 8 0 2 Example continued • Substitute values of x to find y. 3x + 2y = 10 x = 4/3 The solutions are x = 2 • Visualizing the Solution Example • Solve the system of equations: 5 x 2 y 13 2 2 3 x 4 y 39 2 2 Example continued • Solve by elimination. Example continued • Substituting x = 1 in equation (2) gives us: x=1 x = -1 • The possible solutions are Example continued All four pairs check, so they are the solutions. • Visualizing the Solution