Nonlinear Systems of Equations

SAT Math · Axiom Academy

Nonlinear Systems of Equations Find Intersections of Parabolas, Lines, and Circles A system with at least one equation that's not linear (has exponents, curves, etc.). 0, 1, or 2 intersection points Can have 0, 1, 2, 3, or 4 intersections Solve one equation for one variable Substitute into the other equation Solve the resulting equation (usually quadratic) Find corresponding y-values (or vice versa) Solving Line + Parabola Systems Discriminant of the quadratic tells us: Circle: (center at origin, radius r) Determining Number of Solutions Without solving the entire system, the discriminant tells us how many real solutions exist. Common question: "For what value of k does the system have exactly one solution?" When you substitute and get a quadratic in one variable: → 0 solutions (no intersection) → 2 solutions (two intersections) For what value of b does the system have exactly one solution? If asked "how many solutions," use discriminant If asked to "find the solutions," fully solve the system Sketch graphs mentally to visualize the number of intersections

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