Slope (Gradient) of a Line

The slope (or gradient) of a straight line measures its steepness and direction of rise. It is denoted by $m$ and defined through the angle the line makes with the positive direction of the x-axis.

The inclination $\theta$ of a line is the angle, measured anticlockwise, from the positive x-axis to the line, with $0^\circ \le \theta < 180^\circ$. The slope is then

Because $\tan\theta$ is positive for acute $\theta$ and negative for obtuse $\theta$, a line rising left-to-right has positive slope, and a line falling left-to-right has negative slope. A horizontal line has $\theta = 0^\circ$, so $m = 0$. A vertical line has $\theta = 90^\circ$, where $\tan 90^\circ$ is undefined — a vertical line has no slope (its slope does not exist).

When two points on the line are known, the slope can be found without the angle at all. If $A(x_1, y_1)$ and $B(x_2, y_2)$ are any two distinct points on the line,

This is the rise (change in $y$) divided by the run (change in $x$). The order of the two points does not matter, provided the same order is used in the numerator and the denominator: computing $\dfrac{y_1 - y_2}{x_1 - x_2}$ gives exactly the same value. If $x_1 = x_2$ the line is vertical and the slope is undefined, consistent with the inclination view above. …