Point-Slope Form

Pre-Algebra · Axiom Academy

One point and one slope pin down exactly one line — and that single fact is the whole equation. 1. A Point and a Slope Make One Line Pin a known point (x_1, y_1) on the plane, then commit to a slope m — a fixed rise over run. Walk that slope outward in both directions and you trace one line, and only one. Point-slope form packages exactly those two pieces: (x_1, y_1) — the point you know Take any other point (x, y) on the line. The slope between it and your fixed point (x_1, y_1) must equal m — that's what "slope m " means. Watch (x, y) slide along the line: the rise-over-run ratio never budges from 2 . multiply both sides by (x - x_1) to clear the fraction The slope between (x,y) and (x_1,y_1) is , and it equals m . Multiplying both sides by (x - x_1) removes the fraction and gives the point-slope form. With m = 2 and point (3, 4) , the slope condition rearranges directly to y - 4 = 2(x - 3) . Point-slope form is nothing more than the slope formula, solved. Point-slope is built for a point and a slope; slope-intercept is built for the slope and the y -intercept. They describe the same line — to switch, just distribute and solve for y . Here's a line with slope -3 through (2, 5) : Distribute the slope: -3(x - 2) = -3x + 6 , so y - 5 = -3x + 6 . Solve for y : add 5 to both sides to get y = -3x + 11 — the y -intercept is 11 . Reads off a point (2,5) and the slope -3 at a glance. Reads off the slope -3 and the y -intercept 11 at a glance.

This is the written version of the interactive lesson above. See the full Pre-Algebra course.