# 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.

## Resources

### Books

Bittinger, Marvin L,, and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Larson, Ron. Precalculus. 5th ed. New York: Houghton Mifflin College, 2000.