**Theorem
1.3**: There is a uniquely determined
line going through a given
point with a given
slope. If the given
point is (*x*_{1},
*y*_{1}), and the line is
vertical, then the equation is

If the line is not vertical, then the equation can be written as

which can be simplified to

where

**Proof**: If the
point is (*x*_{1},
*y*_{1}) and the line is a vertical
line then its equation is

which all of the points
will satisfy, since by definition, all
points on a
vertical line have the same *x-*coordinate.

If the line is not vertical, then by
Theorem 1.2, it satisfies and equation
of the form *y* = *mx* + *b*. If (*x*, *y*) is any other point on the line, then by
Theorem 1.1, the
slope between it and
(*x*_{1}, *y*_{1}) will be *m*. That is to say.

Clear the denominator

To get this into the form *y* = *mx* + *b*, solve for *y*. Remove
parentheses

Transpose

which is of the form *y* = *mx* + *b* if

*b** = y*_{1}*- mx*_{1}