An undefined slope results from a vertical line, which has multiple x coordinates but no y coordinates. To solve for the slope of a line, it is necessary to select two points and divide the difference in y coordinates from the difference in x coordinates.
In a vertical line, the slope calculation results in a number divided by zero, because by definition, a vertical line has the same x values for every point on the line. Any number divided by zero is by necessity undefined, as there are an infinite number of solutions to the equation.
The slope of a line is also called its rise over its run. When graphed, the rise is the amount that the line goes up or down, and the run is how far right or left the line goes. The calculation for slope is the rise of the line divided by the run of the line. A line with a positive slope goes from lower left to upper right. A line with a negative slope goes from upper left to lower right on a graph. A horizontal line has no slope as there is no rise to it over the length of its run.