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Nonlinear Systems

A free Algebra II lesson from the “Equations, Rational Functions, and Conics” unit, with a worked example and practice problems including step-by-step solutions.

A nonlinear system can include a line with a quadratic, circle, or other curve. Solutions are intersection points that satisfy both equations.

What you'll learn

Why it matters: Where a parabola meets a line, where a circle crosses a curve, and where two growth models intersect are all nonlinear systems. Substitution and graphing still find the intersection — there just may be zero, one, two, or more solutions instead of always exactly one.

Worked example

Problem. Solve y = x + 2 and y = x^2 for x-values.

  1. Set the expressions equal: x + 2 = x^2.
  2. Rearrange: x^2 - x - 2 = 0.
  3. Factor: (x - 2)(x + 1) = 0, so x = 2 or -1.

Answer: -1 and 2

Practice problems

1. Solutions to a system are graphically...

Choices: Intersection points · Only y-intercepts · Only vertices · End behavior

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solutions to a system are graphically...
  2. For a system, use substitution, elimination, or graphing to find the value pair that makes both equations true.
  3. Both equations are true at intersections.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Intersection points

2. Solve y = x and y = 2x - 3. Enter x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve y = x and y = 2x - 3. Enter x.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set x = 2x - 3.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. For y = x^2 and y = 4, enter the positive x-value.

Show solution
  1. Core Practice: First identify exactly what the question is asking: For y = x^2 and y = 4, enter the positive x-value.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x^2 = 4, so x = +/-2.
  4. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

4. A line and parabola can intersect in how many points?

Choices: 0, 1, or 2 · Exactly 3 · Only 0 · Always 4

Show solution
  1. Challenge: First identify exactly what the question is asking: A line and parabola can intersect in how many points?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Depending on the line position.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 0, 1, or 2

5. Substitute y = 2x into y = x^2. Enter the nonzero x-value.

Show solution
  1. Mixed Review: First identify exactly what the question is asking: Substitute y = 2x into y = x^2. Enter the nonzero x-value.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set x^2 = 2x.
  4. Move all terms to get x^2 - 2x = 0.
  5. Factor x(x - 2) = 0, so the nonzero solution is 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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