Line Equation from Two Points Calculator

The linear equation from two points describes the unique straight line that passes through both coordinates. It can be expressed in several equivalent forms: slope-intercept form (y = mx + b), standard form (Ax + By + C = 0), or point-slope form. All forms represent the same line — just written differently.

First, calculate the slope using m = (y₂ − y₁) / (x₂ − x₁). Then find the y-intercept with b = y₁ − m·x₁. Substitute both values into y = mx + b to get the slope-intercept form. For example, for points (−4, 7) and (1, 2), the slope is −1 and the equation is y = −x + 3.

Starting from the slope-intercept form y = mx + b, rearrange to Ax + By + C = 0 by moving all terms to one side. If m = p/q is a fraction, multiply through by q to get integer coefficients. For y = −x + 3, the standard form is x + y − 3 = 0.

The two-point form is (y − y₁) / (y₂ − y₁) = (x − x₁) / (x₂ − x₁). This directly uses both points to define the line without first computing the slope separately. It's equivalent to the point-slope form once the slope is substituted.

The slope is m = (5 − 1) / (3 − 1) = 4 / 2 = 2. The y-intercept is b = 1 − 2·1 = −1. So the slope-intercept form is y = 2x − 1, and the standard form is 2x − y − 1 = 0.

If x₁ = x₂, the denominator of the slope formula becomes zero, meaning the line is vertical. A vertical line cannot be expressed in slope-intercept form — instead, it's written as x = c, where c is the shared x-value. This calculator will alert you when the points form a vertical line.

If both points are the same (x₁ = x₂ and y₁ = y₂), no unique line can be determined — infinitely many lines pass through a single point. The calculator requires two distinct points to compute a valid equation.

Yes. You can enter any real number — including decimals like 2.5 or negative values like −7.3 — as coordinates. The calculator will compute the slope, intercept, and all equation forms with up to four decimal places of precision.