Why is slope undefined for vertical lines?

Answer 1

Because all x-axis values in a vertical line have the same value and this causes the denominator of the slope formula to result in zero when the formula is evaluated/computed. That of course causes the "undefined" division condition to occur.

More Information about the "undefined" condition:

y/x=? asks the question "How many times can x be subtracted from y until a value less than x is the result?

For example: 8/2=?

8-2=6

6-2=4

4-2=2

2-2=0 (this division is possible and definite (i.e. "defined") because it is possible to subtract 2 from 8 exactly 4 times.

But, zero in the denominator causes a problem: 8/0=?

8-0=8

8-0=8

8-0=8

8-0=8 (this division is NOT definite nor definable, hence "undefined" because it is not possible to subtract 0 from 8 "until a value less than 0 is the result." That is the reason dividing by 0 is said to be "undefined".

As stated earlier, when we apply the above knowledge to the slope formula, we can see that a vertical line has the same x values and when substituted into the slope formula "slope = (y2-y1)/(x2-x1)", the denominator will have a result of 0. Having a result of 0 in the denominator causes the "undefined" condition to exist.