The slope of a horizontal line is zero, but why is the slope of a vertical line undefined (not zero)?

It's like the difference between ##0/1## and ##1/0##.

##0/1 = 0## but ##1/0## is undefined.

The ##m## of a line passing through two points ##(x_1, y_1)## and ##(x_2, y_2)## is given by the formula:

##m = (Delta y) / (Delta x) = (y_2 - y_1) / (x_2 - x_1)##

If ##y_1 = y_2## and ##x_1 != x_2## then the line is horizontal: ##Delta y = 0##, ##Delta x != 0## and ##m = 0 / (x_2 - x_1) = 0##

If ##x_1 = x_2## and ##y_1 != y_2## then the line is vertical: ##Delta y != 0##, ##Delta x = 0## and ##m = (y_2 - y_1) / 0## is undefined.