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Chapter 0: Problem 124

Explain why the slope of a vertical line is said to be undefined.

Short Answer

Expert verified
The slope of a vertical line is said to be undefined because the calculation of slope involves division by the 'run' or horizontal distance. For a vertical line, the 'run' is zero. As division by zero is undefined in mathematics, the slope of a vertical line is also considered undefined.

Step by step solution

01

Define Slope

The slope of a line is a measure of how steep that line is. It is usually calculated as the 'rise' over 'run', which in mathematical terms is the change in y (the vertical distance) divided by the change in x (the horizontal distance). Therefore, the slope can be represented as follow: \( m = \frac{\Delta y}{\Delta x} \) where \( \Delta y \) denotes the change in the y-coordinate (rise) and \( \Delta x \) denotes the change in the x-coordinate (run).
02

Understand the Special Case of a Vertical Line

A vertical line goes straight up and down, which means there is no 'run' or horizontal distance. It has a huge 'rise' or vertical distance but a 'run' or horizontal distance of zero. Putting it back into the slope formula we get \( m = \frac{\Delta y}{0} \)
03

Understand the Concept of Division by Zero

In mathematics, division by zero is undefined. This is because zero times any number equals zero, so it is impossible to determine a number that, when multiplied by zero, gives anything other than zero. Therefore, if the denominator in a division operation is zero, the result is undefined.
04

Apply the Principle to the Slope of a Vertical Line

As per the slope formula, the slope of a vertical line involves division by zero which is mathematically undefined. Therefore, the slope of a vertical line is also undefined.

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