Question

Explain what happens when the formula for slope is applied to a vertical line.

Short answer

The slope of a vertical line is undefined because the formula for slope involves a division by the horizontal change, which is zero for a vertical line, and division by zero is undefined.

Step-by-step solution

Understanding the Concept of Slope

The slope of a line measures steepness of that line. It is calculated as the change in y (vertical change) divided by the change in x (horizontal change) between two points on the line. This is often stated as 'rise over run'.

Applying the Slope Formula to a Typical Line

Let's consider a line, where we have two points, say (x1, y1) and (x2, y2). The slope m of this line is given by: \[ m = \frac{y2 - y1}{x2 - x1} \]. This formula gives us a numerical value representing the slope of the line.

Applying the Slope Formula to a Vertical Line

Now, consider a vertical line. In this case, the line goes straight up and down, so the change in x is zero (as the line doesn't run left or right). So, if we try to apply the slope formula, it will look like this: \[ m = \frac{y2 - y1}{0} \] Because we cannot divide by zero, the slope of a vertical line is undefined.