Point-Slope Form

Every remaining form in this chapter is really just a special case, or a short consequence, of the single idea in this section: fixing one point and the slope pins down a line completely, and its equation follows directly from the definition of slope itself.

Setting up. Suppose a (non-vertical) line has slope $m$ and passes through a known, fixed point $(x_1,y_1)$. Let $P(x,y)$ be any other point on the line — $(x,y)$ stands for a completely general point, and the equation sought is the algebraic condition $(x,y)$ must satisfy to lie on this particular line.

Derivation. Since $(x_1,y_1)$ and $(x,y)$ are two points on the same line, the slope computed between them must equal the line's given slope $m$ (Section 2):

Multiplying both sides by $(x-x_1)$ clears the fraction and gives the point-slope form:

This single equation is satisfied by every point on the line and by no point off it, because it says exactly "the slope from $(x_1,y_1)$ to $(x,y)$ is $m$" — true only for points on the line through $(x_1,y_1)$ with that slope.

Why this is the most fundamental form. Every other equation of a line used in this chapter — slope-intercept, two-point, intercept — is obtained from the point-slope form by choosing a convenient point (the y-intercept, one of two given points, an axis intercept) and substituting it in. None of them contains any new geometric idea beyond this one. …