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Methods for Solving Nonlinear Equations

Apr 25, 2025

Lecture Notes: Solving Nonlinear Systems of Equations

Introduction

  • Focus: Solving nonlinear systems of equations.
  • Methods: Graphing, Substitution, Elimination.

Graphing Method

  • Objective: Find the point(s) of intersection of the equations.
  • Tools: Desmos or graphing calculators.
  • Example 1:
    • Equations: (y = 2x^2 + 5x - 1) and (y = x - 3).
    • Intersection at ((-1, -4)).
  • Example 2:
    • Equations: (y = x^2 + 4x - 4) and (y = 2x - 5).
    • Intersection at ((-1, -7)).
  • Example 3:
    • Equations: (y = 3x - 15) and (y = x^2 - 2x - 7).
    • No intersection, i.e., no solution.
  • Multiple solutions possible depending on intersections.

Substitution Method

  • Objective: Substitute one equation into another.
  • Process:
    1. Solve one equation for a variable.
    2. Substitute into the other equation.
    3. Solve the resulting equation to find points.
  • Example:
    • Equations: (y = -2x + 3) and (y = x^2 + x - 1).
    • Solve: (0 = x^2 + 3x - 4) (factor or use the quadratic formula).
    • Solutions: ((1, 1)) and ((-4, 11)).

Elimination Method

  • Objective: Eliminate one variable by adding or subtracting equations.
  • Process:
    1. Align equations.
    2. Add/Subtract to eliminate one variable.
    3. Solve for the remaining variable.
  • Example:
    • Equations: (y = x^2 - 3x - 2) and (y = -3x - 8).
    • Ends with (x^2 = -6), indicating no real solutions.

Graphing Calculator/Desmos

  • Key Tool: Used for visualization and confirming solutions.
  • Examples require finding intersections visually and rounding coordinates.
  • Example 4:
    • Equations: (y = \frac{1}{2}x^2 + 3) and (y = 3^x).
    • Intersection at ((1.194, 3.713)).
  • Example 5:
    • Equations: (y = -2 \times 4^x + 3) and (y = 0.5x^2 - 2x).
    • Intersections at ((-1, 2.5)) and ((0.468, -0.827)).

Conclusion

  • Practice different methods: Graphing, Substitution, Elimination.
  • Graphing is often simplest but check solutions when calculators aren't available.
  • Remember, intersections can lead to no solutions, one solution, or multiple solutions.
  • For questions, seek help during office hours or via Google Meet.

Tips

  • Clearly specify methods used, especially when using Desmos.
  • Always verify solutions, particularly when not using graphing tools.

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