**Theorem 1.2**: A
vertical
line has
infinite
slope. A
line is not
vertical if and only if
its equation can be written in the form

**Proof**: Let (*x*_{1}, *y*_{1}) and
(*x*_{2}, *y*_{2}) be two points on a
vertical
line. Since they are two
points on the same
vertical
line, the by the defintion of
a vertical
line they have the same *x-*coordinates. If they are different points then they must have
different *y-*coordinates. So the slope between them will have
a zero denominator and a nonzero numerator, and will then have a
value of infinity.

If the line satisfies an
equation of the form *y* = *mx* + *b*, we must show that the line is not
vertical. First, since it
will be possible to substitute any real number in for *x*, we could
take any two different real numbers, substitute them in for *x* and get
two points on the
line with different * x* coordinates. So not all points on the
line will have the same *x* coordinates, so the line is not
vertical.

We must finally show that if the
line is not
vertical, then it
satisfies an equation of the form * y* = *mx* + *b*. By definition, a line is the set of
points who's coordinates
satisfy a linear equation in one or two unknowns. Such an equation
can be written as

where *r*, * s*, and *t* are real numbers where *r* and *s* are not both 0.
If *s* = 0, then *r* must not be zero, because, otherwise, our equation
would have no unknowns, so we can solve for *x*. All of the points on the
line would then have the same * x*-coordinate: *x* = *t*/*r*, and the line would be
vertical. So if the
line is not
vertical, then * s* is not
zero. We could then solve for *y*. Transpose

and, since *s* is not 0, divide by *s* and get an equation of the form

where *m* = -*r*/*s* and *b* = *t*/*s*