# Equations of Line

### intercept written slope plane

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 (x_{1}, y_{1}) and (x_{2},y_{2}) 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-y_{1} = m (x-x_{1}) where (x_{1}, y_{1}) 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 (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2}) are points on the line; and the parameter form: x = x_{1} + lt, y = y_{1} + mt, z = z_{1} + nt where the parameter t is the directed **distance** from a fixed point (x_{1},y_{1},z_{1}) 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.

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