Two-point equations are used to solve straight line Equation formulas in a two-dimensional coordinate system, which is an important concept in analytic geometry straight line theory.

The straight line l passes through two points P1(x1,y1)P2(x2,y2)(x1≠x2). So it slope k=(y2-y1)/(x2-x1), substituting into the point-slope equation, we get y=k-(x-x1)+y1, so the two-point equation is (y-y2)/(y1-y2) = (x-x2)/(x1-x2).

Derivation process

If x1=x2, we know that p1p2 and x-axis are perpendicular, then straight line l Equation is x=x1

If y1=y2, we know that p1p2 and y-axis are perpendicular, then the straight line l Equation is y=y1

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Set p(x, y)x, different from p1, p2 's any point, since p, p1, p2 are on straight line l, then kpp1 = kp1p2, both (y-y1)/(x-x1) = (y2-y1)/(x2-x1), collapsed to (y-y1)/(y2-y1) = (x-x1)/(x2-x1).