# How do you find the slope of x=7?

Jun 5, 2018

You can't: it's not defined!

#### Explanation:

The slope is defined as the ratio between the difference of the $y$ components and the difference of the $x$ components of a given pair of points on a line.

In other words, given a line, pick two points ${P}_{1} = \left({x}_{1} , {y}_{1}\right)$ and ${P}_{2} = \left({x}_{2} , {y}_{2}\right)$, the slope $m$ is defined as

$m = \setminus \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

In your case, the line $x = 7$ is composed, as the equation suggests, by all the points having the $x$ component equal to $7$, and any $y$ component. So, two points on the line have the form ${P}_{1} = \left(7 , {y}_{1}\right)$ and ${P}_{2} = \left(7 , {y}_{2}\right)$

Can you see the problem? If we compute the slope, we have

$m = \setminus \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \setminus \frac{{y}_{2} - {y}_{1}}{7 - 7} = \setminus \frac{{y}_{2} - {y}_{1}}{0}$

And you can't divide by zero. This is the reason why all vertical lines (i.e. those with equation $x = k$, for some real number $k$) have no defined slope.