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Mom's Basement University

Nonlinear Systems

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to solve systems that contain both linear and nonlinear equations, focusing on quadratic-linear systems.

The Concept

Section titled “The Concept”

A nonlinear system contains at least one equation that is not linear (usually quadratic, but can be radical, exponential, etc.).

Common case in Algebra 2: one linear equation and one quadratic equation.

Solution methods:

  1. Substitution: solve the linear equation for one variable and substitute into the quadratic.
  2. Elimination (sometimes possible after rearranging).
  3. Graphing: find intersection points visually (useful for checking).

The number of solutions can be 0, 1, or 2 (for quadratic-linear systems). Always check solutions in both original equations.

Worked Example

Section titled “Worked Example”

Solve the system:

{y=x+1y=x2−2x+3\begin{cases} y = x + 1 \\ y = x^2 - 2x + 3 \end{cases}

The graph shows the green line y = x + 1 crossing the blue parabola y = x² − 2x + 3 at two points. The gold dots mark the solutions, the intersection points we need to find algebraically.

Substitution: since both equations equal y, set them equal:

x+1=x2−2x+3x + 1 = x^2 - 2x + 3

Bring all terms to one side:

0=x2−3x+20 = x^2 - 3x + 2

Factor:

(x−1)(x−2)=0⇒x=1   or   x=2(x - 1)(x - 2) = 0 \quad\Rightarrow\quad x = 1 \;\text{ or }\; x = 2

Find the corresponding y-values using y = x + 1:

  • x = 1: y = 1 + 1 = 2 → solution (1, 2)
  • x = 2: y = 2 + 1 = 3 → solution (2, 3)

Check both in the quadratic y = x² − 2x + 3:

  • (1, 2): 1² − 2(1) + 3 = 1 − 2 + 3 = 2 ✓
  • (2, 3): 2² − 2(2) + 3 = 4 − 4 + 3 = 3 ✓

Both solutions are valid.

Real-World Application

Section titled “Real-World Application”

Nonlinear systems model many real situations:

  • Finding intersection points between a straight road (linear) and a curved path (parabolic)
  • Break-even analysis when cost is linear and revenue is quadratic
  • Projectile motion intersecting with a straight line (e.g., hitting a target)
  • Mixing problems with nonlinear relationships

Example: a company has linear cost and quadratic revenue. Solving the system tells them the exact production levels where they break even.

Quiz

Section titled “Quiz”
A system with one linear and one quadratic equation can have up to how many real solutions?
The best first step when solving a linear-quadratic system is usually:
After substitution you usually end up with:
Why is it important to check solutions in nonlinear systems?
Solve the system: $y = x^2$ and $y = x + 2$.