A horizontal line has no slope, but I don't see why a vertical line couldn't either I guess.
But when you have a zero as the denominator, that is undefined, so that should mean that the slope of a vertical line is undefined, so I guess the above explanation wouldn't work.
About the finding the slope of a vertical line, a lot of people (including myself) still believe that because they haven't been taught the necessary information to find the slope of a vertical line.
As far as I know, slope is rise/run. And if the run is zero, then wouldn't the slope be...