Lesson 82: Solving Nonlinear Systems with Three Variables
Nonlinear systems with three variables involve equations in x, y, and z, with at least one equation being nonlinear. Solutions represent points (x, y, z) where the surfaces intersect in 3D space.
These systems appear in physics, engineering, and optimization problems.
Methods for Solving
- Substitution: Solve one equation for a variable and substitute into the other two equations.
- Elimination: Combine equations to eliminate one variable at a time.
- Graphical/Visualization: Use 3D graphs to understand intersections (optional for complex systems).
Example
Solve the system:
x + y + z = 6
x² + y² = 13
z = x − y
Step 1 — Substitute z = x − y into first equation:
x + y + (x − y) = 6 → 2x = 6 → x = 3
Step 2 — Solve for y using second equation:
x² + y² = 13 → 3² + y² = 13 → 9 + y² = 13 → y² = 4 → y = 2 or y = −2
Step 3 — Solve for z:
z = x − y → if y = 2, z = 3 − 2 = 1 → (3, 2, 1)
if y = −2, z = 3 − (−2) = 5 → (3, −2, 5)
Solution: (3, 2, 1) and (3, −2, 5)
Practice Problems
- Solve: x + y + z = 7, x² + y² = 25, z = y − x
- Solve: x − y + z = 4, y² + z² = 10, x = 2y
- Solve: x + 2y + z = 5, x² + z² = 13, y = z − 1
- Solve: x + y + z = 9, x² + y² = 41, z = 2x − y
- Solve: 2x + y − z = 3, x² + y² = 10, z = x + y