Definition:Slope of Straight Line/General Form

Definition

Let $\LL$ be a straight line embedded in a Cartesian plane.

Let $\LL$ be given by the equation:

$l x + m y + n = 0$

The slope of $\LL$ is defined by means of the ordered pair $\tuple {-l, m}$, where:

for $m \ne 0$, $\psi = \map \arctan {-\dfrac l m}$

for $m = 0$, $\psi = \dfrac \pi 2$

where $\psi$ is the angle that $\LL$ makes with the $x$-axis.

Thus:

when $m = 0$, the slope of $\LL$ is $\tuple {l, 0}$ and $\LL$ is parallel to the $y$-axis

when $l = 0$, the slope of $\LL$ is $\tuple {0, m}$ and $\LL$ is parallel to the $x$-axis.

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $9$. Parallel lines. Points at infinity