Nonlinear systems — Digital SAT Math practice
A line with a parabola or circle; counting intersections.
Key terms
- Nonlinear System: A set of equations where at least one graph curves, like a parabola or circle.
- Intersection Point: A coordinate pair that satisfies every equation in the system simultaneously.
- Discriminant: The expression b^2 - 4ac that predicts intersection count without solving fully.
- Substitution Method: Replacing one variable with a linear expression to reduce the system to one equation.
Common traps
- Forgetting to substitute each x-value back into the linear equation to get its y-coordinate.
- Misidentifying a, b and c after moving terms, which breaks the discriminant count.
- Assuming a line and a parabola always intersect twice; the discriminant can be zero or negative.
Key ideas to know
Solving Nonlinear Systems
How do you solve a system with one linear equation and one nonlinear equation?
Substitute the linear expression into the nonlinear equation to create a single quadratic, then solve for x and back-substitute to find y.
Watch out: Forgetting to distribute the substitution across all terms in the nonlinear equation or skipping back-substitution.
Counting Intersections with Discriminant
Without solving, how can you determine if a line and a parabola intersect at two points, one point, or no points?
Find the discriminant of the resulting quadratic after substitution. If b2 - 4ac > 0 there are two intersections; = 0 one intersection; < 0 no intersections.
Watch out: Confusing the discriminant sign with the number of solutions or miscalculating a or b when rearranging to standard form.
Contextual Solution Rejection
You solve a system for t and get t = 2 and t = -5. You write the final answer as t = 2, t = -5 without checking constraints. What is wrong?
The error is accepting both roots without applying the context constraint. Since time cannot be negative, reject t = -5 and keep only t = 2.
Watch out: Assuming every algebraic solution is valid regardless of physical constraints or domain limits like square roots or denominators.
Sample questions
Question 1
In the xy-plane, at how many points do the graphs of y = x² − 4 and y = 2x − 1 intersect?
- 0
- 1
- 2
- 3
Show answer and explanation
Answer: 2
Set equal: x² − 4 = 2x − 1 → x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0. Two solutions (x = 3, x = −1), so two intersection points.