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College Algebra

Nonlinear Systems

Lesson

A nonlinear systemhas at least one curve that isn’t a line — a parabola, a circle, an exponential. The big tool is the same as for linear systems: substitution.

The strategy

  1. Solve the simpler equation for one variable (usually y).
  2. Substitute into the other equation.
  3. Solve the resulting one-variable equation (often quadratic).
  4. Plug each solution back to find the matching y.

Worked example 1 — parabola meets line

y=x2,y=x+2y = x^2, \quad y = x + 2

Set them equal:

x2=x+2  ⟹  x2−x−2=0x^2 = x + 2 \implies x^2 - x - 2 = 0
(x−2)(x+1)=0  ⟹  x=2 or x=−1(x - 2)(x + 1) = 0 \implies x = 2 \text{ or } x = -1

Worked example 2 — circle meets line

x2+y2=25,y=0x^2 + y^2 = 25, \quad y = 0

Substitute:

x2+0=25  ⟹  x=±5x^2 + 0 = 25 \implies x = \pm 5

How many solutions?

  • A line and a parabola: 0, 1, or 2 intersection points.
  • A line and a circle: 0, 1, or 2 as well.
  • Two parabolas: up to 4 (rare in intro problems).

How to type your answer

List the x-coordinates of every intersection, separated by commas. Example: 2,-1. Order doesn’t matter.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

{y=x2y=9\begin{cases} y = x^2 \\ y = 9 \end{cases}

Problem 2

{y=x2y=16\begin{cases} y = x^2 \\ y = 16 \end{cases}

Problem 3

{y=x2+1y=10\begin{cases} y = x^2 + 1 \\ y = 10 \end{cases}

Problem 4

{y=xy=x2\begin{cases} y = x \\ y = x^2 \end{cases}

Practice

Standard problems matching the lesson.

Problem 5

{y=x2y=25\begin{cases} y = x^2 \\ y = 25 \end{cases}

Problem 6

{y=x2−4y=0\begin{cases} y = x^2 - 4 \\ y = 0 \end{cases}

Problem 7

{y=x2y=2x\begin{cases} y = x^2 \\ y = 2x \end{cases}

Problem 8

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

Problem 9

{y=x2y=x+2\begin{cases} y = x^2 \\ y = x + 2 \end{cases}

Problem 10

{y=x2+xy=6\begin{cases} y = x^2 + x \\ y = 6 \end{cases}

Problem 11

{y=x2−3xy=4\begin{cases} y = x^2 - 3x \\ y = 4 \end{cases}

Problem 12

{y=x2y=3x−2\begin{cases} y = x^2 \\ y = 3x - 2 \end{cases}

Problem 13

{y=x2−6y=x\begin{cases} y = x^2 - 6 \\ y = x \end{cases}

Problem 14

{y=x2+2xy=8\begin{cases} y = x^2 + 2x \\ y = 8 \end{cases}

Problem 15

{x2+y2=25y=0\begin{cases} x^2 + y^2 = 25 \\ y = 0 \end{cases}

Problem 16

{x2+y2=13y=2\begin{cases} x^2 + y^2 = 13 \\ y = 2 \end{cases}

Problem 17

y = x^2 meets y = x + 12. x?

Problem 18

x^2 + y^2 = 10 meets y = 1. x?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

{y=x2−2x+1y=4\begin{cases} y = x^2 - 2x + 1 \\ y = 4 \end{cases}

Problem 20

{y=2x2y=8\begin{cases} y = 2x^2 \\ y = 8 \end{cases}

Problem 21

{y=x2+4y=5x\begin{cases} y = x^2 + 4 \\ y = 5x \end{cases}

Problem 22

{y=−x2+4y=3\begin{cases} y = -x^2 + 4 \\ y = 3 \end{cases}

Problem 23

{y=x2−5y=2x−5\begin{cases} y = x^2 - 5 \\ y = 2x - 5 \end{cases}

Problem 24

{x2+y2=50y=5\begin{cases} x^2 + y^2 = 50 \\ y = 5 \end{cases}

Problem 25

{y=x2y=−x+6\begin{cases} y = x^2 \\ y = -x + 6 \end{cases}

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