Prove that two lines are parallel if and only if they have the same slope

Given line q and line r, with equations

such that these are different lines, with identical slopes. Prove that line q || line r.

If these lines are parallel, then they will not intersect. Assume the lines intersect. Calculate the point of intersection. Since these equations already have the same slope, we may as well subtract one equation from the other one.

This gives us the following equation:

If b1 = b2, then line q is the same line as line r. This contradicts the given conditions.

Therefore, line q and line r cannot intersect. If line q and line r do not intersect, they must be parallel.