**Theorem
1.6**: If two distinct
lines are
parallel, then they do
not meet. If they are not
parallel, then they
meet at a uniquely determined
point. If one of the
lines is
vertical, its equation will be of the
form

If the other one is not vertical, it will have an equation of the form

*y* = *mx* + *b*

and the point of intersection will be

If neither of the lines are vertical, they will have equations of the form

y

The coordinates of the point of intersection are

and

**Proof**: If both
lines are
vertical then by
Theorem 1.2, they both have
infinite
slope, so, by definition,
they will be parallel.
By Theorem 1.3, their equations
will be

If the lines are
distinct, then *a*_{1} and *a*_{2} will be different, and there will be no solution.

If one is vertical, its equation will be

and the other is not, its equation will be of the form

by Theorem 1.2. They have different slopes, so they are not parallel. The unique point of intersection is

Otherwise, neither line is vertical, and their equations will be of the form

y

again, by Theorem 1.2. This system is set up for the comparison method. Set the right sides equal.

Transpose

Combine *x* terms

Divide

and the *x* coordinate is uniquely determined unless the denominator
is zero. That will happen if and only if the slopes are the same which,
by definition is when the
lines are
parallel.

To find the *y* coordinate of the point of intersection, we could
substitute this solution for *x* into either of the original equations.
If we substitute it into the first equation, we get

Find common denominators