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Formulas/maths/Straight Lines/Point–Slope Form

Point–Slope Form

Line with slope m passing through (x₁, y₁).
Derivation

A line with slope mm passes through a fixed point P(x1,y1)P(x_1, y_1). Let Q(x,y)Q(x, y) be any other point on it.

By definition of slope:

y−y1x−x1=m\frac{y - y_1}{x - x_1} = m

Multiply both sides by (x−x1)(x - x_1):

y−y1=m(x−x1)\boxed{y - y_1 = m(x - x_1)}

This holds for every point (x,y)(x, y) on the line. The base point (x1,y1)(x_1, y_1) is fixed; (x,y)(x, y) varies.

Note
This is the foundational form — slope-intercept, two-point, and intercept forms all follow from it by substituting specific choices of point or slope.