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: $\left(x-1\right)\left(x-2\right) = 0$, 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.