Algebra 101 - Unit 13

Lesson 80: Solving Nonlinear Systems Algebraically

Nonlinear systems often require algebraic methods such as substitution or elimination to find exact solutions. The goal is to reduce the system to a single equation with one variable.

Steps for Solving by Substitution

1. Solve one equation for one variable in terms of the other.
2. Substitute this expression into the other equation.
3. Solve the resulting equation (quadratic or other nonlinear equation).
4. Find the other variable by substituting back.
5. Check all solutions in the original system.

Examples

Example 1 — Quadratic and Linear:

Solve:
y = x²
y = 2x + 3

Substitute y = x² into y = 2x + 3 → x² = 2x + 3 → x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0 → x = 3, −1
Corresponding y-values: y = 9, 1 → Solutions: (3, 9), (−1, 1)

Example 2 — Circle and Line:

Solve:
x² + y² = 9
y = x + 1

Substitute y = x + 1 → x² + (x + 1)² = 9 → x² + x² + 2x + 1 = 9 → 2x² + 2x − 8 = 0 → x² + x − 4 = 0
Solve: x = [−1 ± √(1 + 16)]/2 = [−1 ± √17]/2 → y = x + 1 → Solutions: ((−1 + √17)/2, (1 + √17)/2), ((−1 − √17)/2, (−1 + √17)/2)

Practice Problems

1. Solve: y = x² − 1, y = 2x + 3
2. Solve: x² + y² = 16, y = 3x − 2
3. Solve: y = −x² + 4, y = x + 1
4. Solve: (x − 2)² + (y + 1)² = 25, y = 2x − 3
5. Solve: y = x², y = −x + 2