ALEKS Practice Exam 2025 – The Complete All-in-One Guide to Mastering Assessment and Learning in Knowledge Spaces!

A line has an undefined slope when it is vertical. This is because slope is defined as the ratio of the change in y to the change in x (rise over run). In the case of a vertical line, there is no change in the x-coordinates, which means that the denominator in the slope formula (change in x) is zero. As a result, the slope cannot be calculated, leading to an undefined value.

Vertical lines run straight up and down, which visually indicates that they do not increase or decrease horizontally at all. The formula used to find the slope (\(m = \frac{\Delta y}{\Delta x}\)) becomes problematic here since you cannot divide by zero.

In contrast, horizontal lines have a slope of zero because the change in y coordinates is zero, and thus they represent no vertical change. Lines that intersect the x-axis do not necessarily relate to the concept of slope being undefined; they can be horizontal, vertical, or sloped. Therefore, the only situation where a slope is truly undefined is when a line is vertical.