Assessment and Learning in Knowledge Spaces (ALEKS) Practice Exam

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When does a line have an undefined slope?

When it has the same y points but different x points
When it is horizontal
When it is vertical
When it intersects the x-axis

The correct answer is: When it is vertical

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.