Solving a Nonlinear System

Solving a nonlinear system by substitution means setting the expressions equal or substituting one into the other, solving the resulting single-variable equation, and checking every candidate pair in both originals.

Solve y = x² with y = x + 2. Both name y, so set them equal: x² = x + 2. Standard form: x² − x − 2 = 0, which factors as (x − 2)(x + 1) — candidates x = 2 and x = −1. Each carries its partner: y = 4 and y = 1. Check both pairs in both equations: (2, 4) gives 4 = 4 twice; (−1, 1) gives 1 = 1 twice. Two intersections, as the picture promised — and the quadratic middle is Video 198's machinery unchanged. The candidate count comes from the degree; the survivors come from the check.