# If x= -3, what is the slope, and what is the y-intercept?

Jul 27, 2015

The slope is $\infty$ and the $y$-intercept does not exist.

#### Explanation:

Recall that the equation for a line can be expressed in the form

$y = m x + b$ where $m = \frac{{x}_{2} - {x}_{2}}{{y}_{2} - {y}_{1}} = \left(\text{vertical rise")/("horizontal run}\right)$

Your $y$-intercept is the value of $y$ when $x = 0$. When you plug a zero into the above equation you get the following:

$y = m x + b$
$y = m \left(0\right) + b$, where we let $x = 0$
$y = 0 + b$, because anything times zero is zero
$y = b$

So the $y$-intercept is the value of $y$ when you set $x = 0$. But you were given a problem where $x$ can never be $0$. You were given that $x = - 3$ and, clearly, $- 3 \ne 0$. Well, if $x \ne 0$ (because it's -3), then the $y$-intercept does not exist.

What about the slope? Remember the slope is $\left(\text{rise")/("run}\right)$. If you want to know how "steep" a line is, you must divide your vertical distance by your horizontal distance. A typical line (not your question above) looks like this:

In order to turn this into a vertical line, you would have to make the run part really short, and the rise part really big. The graph starts to look like this:

So when you are given $x = - 3$, you have a vertical line. A vertical line has zero run because there is simply no left or right steepness. Additionally, the rise becomes infinite because there is only up and down in a vertical line. It's all rise and no run! Thus, the slope, $m = \infty$.

Jul 27, 2015

I would answer that neither the slope nor the $y$ intercept exist.

#### Explanation:

Slope is defined for two points with different $x$ coordinates. This line contains no two such points. Hence the slope is not defined.

A $y$ intercept occurs when we make $x$ equal to $0$. Since this is prohibited by the equation $x = - 3$, there is no $y$ intercept.