Lesson 79: Introduction to Nonlinear Systems
A nonlinear system is a system of equations in which at least one equation is not linear. Nonlinear equations can involve quadratics, circles, parabolas, ellipses, or other curves.
Unlike linear systems, nonlinear systems can have zero, one, two, or more solutions depending on the intersection of curves.
Examples of Nonlinear Systems
Example 1 — Quadratic and Linear:
y = x²
y = 2x + 3
Graphing will show the parabola and the line intersecting at two points → two solutions.
Example 2 — Circle and Line:
x² + y² = 9
y = x + 1
The circle of radius 3 intersects the line at two points → two solutions.
Example 3 — Two Circles:
x² + y² = 16
(x − 3)² + y² = 9
Two circles intersect at two points → two solutions.
Methods for Solving Nonlinear Systems
- Graphing: Plot all equations and find intersection points.
- Substitution: Solve one equation for a variable and substitute into the other.
- Elimination (less common): Can be used if equations can be manipulated to eliminate a variable.
Most nonlinear systems require careful algebraic manipulation and checking of solutions.
Practice Problems
- Solve by graphing: y = x² − 4, y = x + 2
- Find the solution algebraically: y = x² + 1, y = 3x − 1
- Solve: x² + y² = 25, y = x − 1
- Determine the intersection points: y = −x² + 4, y = 2x + 1
- Solve: (x − 1)² + (y + 2)² = 16, y = x