Lesson: Linear and Nonlinear Systems
A system of equations is a set of two or more equations with the same variables. Systems can be classified as linear or nonlinear:
- Linear System: Each equation represents a line. Standard form: Ax + By = C.
- Nonlinear System: At least one equation is not linear (e.g., quadratic, circle, ellipse, parabola).
Methods for Solving Systems
Common methods include:
- Graphing: Plot the equations and find intersection points.
- Substitution: Solve one equation for a variable and substitute into the other.
- Elimination: Combine equations to eliminate a variable.
- Special for nonlinear systems: Sometimes requires substitution or factoring to reduce to a simpler equation.
Examples
Example 1 (Linear System):
System: x + y = 5 2x − y = 1
Solution (Substitution): y = 5 − x → 2x − (5 − x) = 1 → 2x − 5 + x = 1 → 3x = 6 → x = 2 → y = 3
Example 2 (Nonlinear System):
System: y = x² x + y = 6
Solution (Substitution): x + x² = 6 → x² + x − 6 = 0 → (x + 3)(x − 2) = 0 → x = −3, 2 Corresponding y-values: y = 9, 4 → Solutions: (−3, 9), (2, 4)
Practice Problems
- Solve the linear system: 3x + 2y = 12, x − y = 1
- Solve the nonlinear system: y = x² − 1, y = 2x + 3
- Solve the linear system using elimination: 2x + 3y = 7, 4x − y = 5
- Solve the nonlinear system: x² + y² = 25, y = x + 1
- Determine if the system has one, two, or no solutions: y = 2x − 1, y = 2x + 3