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

x = a

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

(a, ma + b)

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

y = m1x + b1
y
= m2x + b2

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

x = a1
x = a2

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

x = a

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

y = mx + b

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

(a, ma + b)

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

y = m1x + b1
y
= m2x + b2

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

m2x + b2 = m1x + b1

Transpose

m2x - m1x = b1 - b2

Combine   x   terms

(m2 - m1)x = b1 - b2

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

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