Equation of a Line
Find the equation of a line through two points, from a point and a vector or the slope: general, slope-intercept, symmetric, parametric and intercept forms with a graph.
The line equation calculator writes the equation of a line through two points, from a point and a direction vector, a point and the slope, a point and a normal vector, and converts the general equation Ax + By + C = 0 into all other forms. In the plane it plots the line; in space it gives the symmetric and parametric equations.
Forms of the equation of a line in the plane
| Form | Equation |
|---|---|
| General (standard) | Ax + By + C = 0, normal vector n = (A, B) |
| Slope-intercept | y = kx + b, k = tan α |
| Two-point | (x − x1)/(x2 − x1) = (y − y1)/(y2 − y1) |
| Symmetric | (x − x0)/l = (y − y0)/m, direction vector (l, m) |
| Parametric | x = x0 + lt, y = y0 + mt |
| Intercept | x/a + y/b = 1 |
Example 1. The line through A(1, 2) and B(4, 8).
Direction vector (3, 6) ~ (1, 2): (x − 1)/1 = (y − 2)/2 ⇒ 2x − y = 0, y = 2x.
Example 2. 2x − 3y + 6 = 0.
y = ⅔x + 2, intercept form: x/(−3) + y/2 = 1, inclination ≈ 33.69°.
A line in space
It is given by a point M0(x0, y0, z0) and a direction vector (l, m, n):
(x − x0)/l = (y − y0)/m = (z − z0)/n, x = x0 + lt, y = y0 + mt, z = z0 + nt
A zero in a denominator of the symmetric equation means that coordinate is constant.
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