Point-Slope Form

When you know a slope and any one point, point-slope form gets you an equation in a single substitution.

For slope 3 through (2, −5):
y − (−5) = 3(x − 2), which tidies to y + 5 = 3(x − 2).

Then expand if you want slope-intercept form: y + 5 = 3x − 6, so y = 3x − 11.

Note both subtractions in the formula. A point with negative coordinates produces a double negative, which becomes a plus.

Given two points instead, compute the slope first, then use either point. Both give the same final line.

Point-slope form is the slope formula rearranged. Starting from m = (y − y₁)/(x − x₁) for a general point (x, y) on the line and multiplying through by (x − x₁) gives y − y₁ = m(x − x₁), valid for every point on the line including (x₁, y₁) itself.

It is the most efficient form for construction because it requires no solving — the given data substitutes directly — while slope-intercept form requires first finding b.

Either of two given points may be used; the resulting equations look different but are equivalent, as expanding both to slope-intercept form confirms. This is a useful check on a written answer that does not match a key.

The same structure recurs throughout mathematics: the tangent line in calculus is y − f(a) = f′(a)(x − a), which is point-slope with the derivative supplying the slope.