Chapter 10 Conic Sections 10.1 – The Parabola and the Circle 10.2 – The Ellipse 10.3 – The Hyperbola 10.4 – Nonlinear Systems of Equations and Their Applications

Summary: Distance & Midpoint Formulas Circles Parabolas Ellipses Area of the ellipse: 𝐴=𝜋𝑎𝑏

10.4 Nonlinear Systems of Equations and Their Applications 1. Solve nonlinear systems of equations using substitution. 2. Solve nonlinear systems of equations using elimination.

Nonlinear System of Equations A nonlinear system of equations is a system of equations in which at least one equation is not linear, that is, one whose graph is not a straight line. No points of intersection: no solutions One point of intersection: One solution Two points of intersection: two solutions

Solve Nonlinear Systems Using Substitution Example 1: Solve x2 + y2 = 20, (1) (a circle.) y – 2x = 0. (2) (a line.) Solution Solve the second equation for y: y = 2x. (3) x2 + (2x)2 = 20 x2 + 4x2 = 20 Now substitute these numbers for x in equation (3) and solve for y: 5x2 = 20 x2 = 4 The solutions are: (2, 4) & (–2, –4).

Solve Nonlinear Systems Using Elimination (Addition) Example 2: Solve. 3x2 + 2y2 = 66, (1) x2 – y2 = 7. (2) 3x2 + 2y2 = 66 2x2 – 2y2 = 14 ( )2 – y2 = 7 5x2 = 80 16 – y2 = 7 x2 = 16 –y2 = – 9 x = y2 = 9 y = . The system has four solutions: (4, 3), (4, –3), (–4, 3), and (–4, –3).

Solve Nonlinear Systems Using Elimination (Addition) Example 3: Solve the system of equations. Solution The solutions are

Solve Nonlinear Systems Using Substitution Example 4: Solve the system of equations.

Solve Applications of Nonlinear Systems Example 5: Given the rectangular area of 48 sq ft along a riverbank as illustrated; find the dimensions, if 20 ft of fencing is needed.

Solve Applications of Nonlinear Systems Example 6: Q48 C=15 a b

Solve Nonlinear Systems of inequalities Example 6: raph the solution set of the system of inequalities. Solution We begin by graphing The intersection of the two regions is the solution set.