# What is the slope of a line that is parallel to a vertical line?

Jul 22, 2015

Any line that is parallel to a vertical line is also vertical and has undefined slope.

#### Explanation:

A vertical line is given by the equation $x = a$ for some constant $a$. This line passes through the points $\left(a , 0\right)$ and $\left(a , 1\right)$.

Its slope $m$ is given by the formula:

$m = \frac{\Delta y}{\Delta x} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{1 - 0}{a - a} = \frac{1}{0}$

which is undefined.

Jul 22, 2015

A vertical line and all lines parallel to it have undefined slopes

#### Explanation:

Note that if a line is vertical, all lines parallel to it are also vertical.

For any two points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ on a line
the slope is defined as $\frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

BUT if the line is vertical ${x}_{1} = {x}_{2}$ for all points on the line
and therefore the definition of the slope would require dividing by zero (which is undefined).