Equations of Line

There are many different ways of writing the equation of a line in a coordinate plane. They all stem from the form ax + by + c = 0. Thus 2x + 3y - 5 = 0 is an equation of a line, with a = 2, b = 3, and c = -5. When the equation is written in the form y = mx + b we have slope-intercept form: m is the slope of the line and b is the y-intercept. The equation 2x + 3y - 5 = 0 becomes

So the line has slope -2/3 and a y-intercept 5/3.

When the equation is written in the form

we have the intercept form: a is the x-intercept and b is the y-intercept. The equation 2x + 3y -5 = 0 becomes

with x-intercept 5/2 and y-intercept 5/3.

When the equation is written in the form

where (x₁, y₁) and (x₂,y₂) are points on the line, we have the two point form. If we choose the two points (1, 1) and (-2, 3) that lie on the line 2x + 3y-5 = 0, we have

When the equation is written in the form y-y₁ = m (x-x₁) where (x₁, y₁) is a point on the line, we have the point-slope form. If we choose (-2, 3) as the point that lies on the line 2x + 3y = 0, we have y - 3 = -2/3 (x + 2).

In three space, a line is defined as the intersection of two non-parallel planes, such as 2x + y + 4z = 0 and x + 3y + 2z = 0. Standard equations of a line in three space are the two-point form:

where (x₁,y₁,z₁) and (x₂,y₂,z₂) are points on the line; and the parameter form: x = x₁ + lt, y = y₁ + mt, z = z₁ + nt where the parameter t is the directed distance from a fixed point (x₁,y₁,z₁) on the plane to any other point (x,y,z) of the plane, and l, m, and n are any constants.