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
The coordinates of the point of intersection are
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 a1 and a2 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
again, by Theorem 1.2. This system is set up for the comparison method. Set the right sides equal.
Combine x terms
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
next theorem (1.7)