# Line, Equations of

# Line, Equations of

In mathematics, a line is a straight one-dimensional structure that has no thickness and extends in both directions, usually without end. An equation of a line is a mathematical statement about some aspect of a 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 one has the 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

one has 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, one has the two point form. If one chooses the two points (1, 1) and (-2, 3) that lie on the line 2x + 3y-5 = 0, one 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, one has the point-slope form. If one chooses (-2, 3) as the point that lies on the line 2x + 3y = 0, one has 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.

Burton, David M. *The History of Mathematics: An Introduction.* New York: McGraw-Hill, 2007.

Lial, Margaret L. *Precalculus.* Boston, MA: Addison-Wesley, 2001.

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