Two Crossings

A line and a parabola can meet. The meeting points are the solutions of a system: pairs (x, y) that make both equations true at once. Take y = x^2 + 1 and y = 3x - 1.

Both equations say what y is, so at a meeting point the two right sides are equal. Substitute one into the other: x^2 + 1 = 3x - 1, a quadratic in x alone. Move every term to one side, x^2 - 3x + 2 = 0, and factor: (x - 1)(x - 2) = 0, so x is 1 or 2.

Each x needs its own y. Put it into the line, the simpler equation: x = 1 gives y = 2, and x = 2 gives y = 5. The meeting points are (1, 2) and (2, 5). An answer is a pair, so an x without its y is only half of one.