Algebra 2 · Unit 8 · Topic 8.1
Intersections of Lines and Curves
A nonlinear system may have several solutions, one solution, or none.
Definition and explanation
A nonlinear system may have several solutions, one solution, or none. Substitution can reduce a line-curve intersection to a polynomial equation; each real candidate must satisfy both original equations.
Worked example
Solve y = x + 2 and y = x². Equating gives x² − x − 2 = 0, or (x − 2)(x + 1) = 0. The intersections are (2, 4) and (−1, 1), both satisfying the line and parabola.
Common mistake
Finding an x-value alone is not a complete ordered-pair solution.
Key takeaway
A nonlinear system may have several solutions, one solution, or none.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.