Equation of Straight Line in Plane/Slope-Intercept Form

Theorem
Let $\mathcal L$ be the straight line defined by the general equation:


 * $\alpha_1 x + \alpha_2 y = \beta$

Then $\mathcal L$ can be described by the equation:
 * $y = m x + c$

where:

such that $m$ is the slope of $\mathcal L$ and $c$ is the $y$-intercept.

Proof
Setting $x = 0$ we obtain:


 * $y = \dfrac {\beta} {\alpha_2}$

which is the $y$-intercept.

Differentiating $(1)$ $x$ gives:


 * $y' = -\dfrac {\alpha_1} {\alpha_2}$

By definition, this is the slope of $\mathcal L$ and is seen to be constant.

The result follows by setting:

Also presented as
This equation can also be seen presented as:


 * $y = x \tan \psi + c$

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