Line Equation from Two Points Calculator

A straight line in the Cartesian plane is completely determined by any two distinct points on it. Given points (x₁, y₁) and (x₂, y₂), this calculator computes the line's equation in three standard forms and provides the slope, y-intercept, and the distance between the two input points. The slope m of a line is the ratio of the vertical change to the horizontal change between any two points on the line: m = (y₂ − y₁) / (x₂ − x₁). The slope tells you how steeply the line rises or falls. A positive slope means the line rises from left to right; a negative slope means it falls. A slope of zero gives a horizontal line; an undefined slope (division by zero, when x₁ = x₂) gives a vertical line. The slope-intercept form y = mx + b is the most commonly used representation. It expresses y as a linear function of x, where m is the slope and b is the y-intercept — the value of y when x = 0. To find b, substitute one of the known points and the computed slope: b = y₁ − m · x₁. The point-slope form y − y₁ = m(x − x₁) is convenient when you know the slope and one point but do not need to find the y-intercept explicitly. It is often the form encountered in differential equations and tangent-line problems in calculus. The standard form Ax + By = C is preferred in many algebraic contexts and for systems of linear equations. In this form, A, B, and C are integers with A ≥ 0 and GCD(|A|, |B|, |C|) = 1. Converting from slope-intercept form: multiply through by the denominator of m if m is fractional, then rearrange to move x and y to the left side. The Euclidean distance between the two input points is computed as √[(x₂ − x₁)² + (y₂ − y₁)²], which follows directly from the Pythagorean theorem applied to the right triangle formed by the two points and their horizontal and vertical legs. Special cases: a horizontal line (y₁ = y₂) has slope 0 and equation y = y₁. A vertical line (x₁ = x₂) has undefined slope and equation x = x₁ — it cannot be written in slope-intercept form. This calculator handles both cases and clearly labels the result. This tool is useful for analytic geometry, linear algebra, physics (projectile paths, kinematics), machine learning (drawing decision boundaries), data analysis (trend lines), and everyday tasks like map navigation, carpentry angles, and road gradients.