# What is the slope for x = 4?

Jul 27, 2015

The slope is not defined for points with the same $x$ coordinate.

#### Explanation:

The definition of slope is for the slope of a line through the points $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$ with ${x}_{1} \ne {x}_{2}$.

The case ${x}_{1} = {x}_{2}$. is not defined.

(You may often hear people say that the slope is infinity. This is the result of a confusion of two or more ideas.)

Jul 27, 2015

A vertical line has an infinitely steep slope because it's straight up and down!

#### Explanation:

Remember that the a typical equation of a line can be expressed as

$y = m x + b$

where $m$ is the slope of the line. The slope of a line describes ratio of rise (the difference in vertical distance, or $y$-values), divided by the run (the difference in horizontal distance, or $x$-values). In other words, slope can be defined as:

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

What this means is that as the top part of the fraction gets big (compared to the denominator), the slope gets steeper and steeper, ever creeping closer to a vertical line. Here you have a slope of just 5:

graph{5x+1 [-11.25, 11.26, -5.63, 5.62]}

And here's a slope of 50:

graph{50x+1 [-11.25, 11.26, -5.63, 5.62]}

So the line becomes vertical as $m$ gets large. But the equation $x = 1$ is simply a vertical line at $x = 1$. So the slope is $\infty$.