1. 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

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

3. 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.

4. 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

5. Solve Nonlinear Systems Using Substitution
Example 1: Solve
x2 + y2 = 20, (1) (a circle.)
y – 2x = (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).

6. Solve Nonlinear Systems Using Elimination (Addition)
Example 2: Solve.
3x2 + 2y2 = 66, (1)
x2 – y2 = (2)
3x2 + 2y2 = 66
2x2 – 2y2 = 14
( )2 – y2 = 7
5x = 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).

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

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

9. 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.

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

11. 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.