# Slope

The slope of a line can be intuitively defined as how steep the line is, relative to some coordinate system. To be more precise, given a line $\mathcal L$ in the Cartesian plane and two points, $(x_1,y_1)$ and $(x_2,y_2)$, on $\mathcal L$ with $x_1 \neq x_2$, the slope $m$ of $\mathcal L$ is equal to $\frac{y_1-y_2}{x_1-x_2}.$ If all points on $\mathcal L$ have the same $x$-coordinate (abscissa), we say that $\mathcal L$ has infinite slope.

Other expressions for the slope are

 $\frac{\rm{rise}}{\rm{run}},$ $\frac{\Delta y}{\Delta x}$ (read "delta $y$ over delta $x$"), or $\frac{\rm{change \ in \ } y}{\rm{change \ in \ } x}$.

If $\theta$ is the directed angle between the $x$-axis and $\mathcal L$, the slope is also given by $m = \tan \theta$.