Two-Point Form

Frequently a line is described not by a point and a slope but by two points it passes through; this section shows the slope-then-point-slope route that handles this case in two short steps.

Setting up. Let $(x_1,y_1)$ and $(x_2,y_2)$ be two distinct given points on a line, with $x_1\ne x_2$ (so the line is not parallel to the y-axis — that case is handled directly by Section 4).

Step 1 — find the slope. By the slope formula of Section 2,

Step 2 — substitute into the point-slope form. Using $(x_1,y_1)$ as the fixed point in the point-slope form of Section 5,

This is the two-point form. It is equally written in the more symmetric layout

by dividing both sides by $y_2-y_1$ (when $y_2\ne y_1$) — a form making the symmetry between the two given points visually clear, since swapping the labels $1\leftrightarrow2$ throughout leaves the equation unchanged.

Why the choice of which point to substitute doesn't matter. Using $(x_2,y_2)$ instead of $(x_1,y_1)$ as the fixed point in Step 2 gives $y-y_2=m(x-x_2)$; expanding both versions shows they simplify to the same general equation $Ax+By+C=0$, since both describe the one line already pinned down uniquely by two points. …