Point-Slope Form of a Line

Point-slope form is the natural way to write a line when you know one point on it and how steep it is, which in practice is nearly always. It is the slope definition m = (y − y₁)/(x − x₁) with the denominator multiplied across, then solved for y so the calculator can evaluate it. Worked example: the line through (3, 4) with slope −2 reaches y = 4 + (−2)(7 − 3) = −4 at x = 7.

Compared with slope-intercept form, point-slope needs no detour through the y-intercept: handy when the intercept is far off the page or physically meaningless, as with a thermometer calibration that passes through (100 °C, 212 °F) with slope 1.8. Two traps recur. The subtraction is x − x₁, so a known point at x₁ = −5 gives (x + 5); and the point you substitute must actually lie on the line, or every value that follows is wrong. Tangent lines in calculus are written in exactly this form for the same reason: differentiation hands you a point and a slope, never an intercept.