Lesson 17.5

Systems of Nonlinear Equations

Finding where curves intersect—lines meet circles, parabolas cross each other.

Introduction

When at least one equation involves , , or , we have a nonlinear system. Instead of intersecting lines, we're finding where curves meet—a circle and a line, two parabolas, or even two circles. The solution could be 0, 1, 2, or even 4 points depending on how the curves interact.

1

Prerequisite Connection

You can solve linear systems and equations like .

2

Today's Increment

We use substitution to find intersections of curves (circles, parabolas).

3

Why This Matters

Finding intersections is central to optimization and multivariable calculus.

Key Concepts

What Makes It Nonlinear?

At least one equation contains , , , or other nonlinear terms.

Substitution Method (Most Common)

1

Solve the LINEAR equation for one variable (if there is one)

2

Substitute into the nonlinear equation

3

Solve the resulting equation (often quadratic)

4

Back-substitute to find all pairs

Line & Circle

0, 1, or 2 solutions

Line & Parabola

0, 1, or 2 solutions

Two Conics

0 to 4 solutions

Worked Examples

Example 1: Line and Circle (Basic)

Find the intersections:

Step 1: Substitute

Step 2: Simplify

Step 3: Solve

Solutions: and

Example 2: Line and Parabola (Intermediate)

Find the intersections:

Set equations equal

Solve quadratic

Find y-values

:

:

Solutions: and

Example 3: Two Circles (Advanced)

Find the intersections:

Strategy: Subtract to eliminate

Simplify

Find y

Solutions: and

Common Pitfalls

Missing ± solutions

When solving , remember .

Not verifying solutions

Always check each solution in BOTH original equations.

Ignoring no-solution cases

If the discriminant is negative or you get no real roots, the curves don't intersect.

Real-World Application

GPS Positioning (Trilateration)

GPS satellites broadcast their position and time. Your device calculates distance to each satellite, forming spheres of possible locations. Finding where three spheres intersect (a nonlinear system!) pinpoints your location.

In 2D, this simplifies to finding where circles intersect—exactly what we practiced today.

Practice Quiz

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