# 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:

 $\displaystyle m$ $=$ $\displaystyle -\dfrac {\alpha_1} {\alpha_2}$ $\displaystyle c$ $=$ $\displaystyle \dfrac {\beta} {\alpha_2}$

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

## Proof

 $\displaystyle \alpha_1 x + \alpha_2 y$ $=$ $\displaystyle \beta$ $\displaystyle \leadsto \ \$ $\displaystyle \alpha_2 y$ $=$ $\displaystyle y_1 - \alpha_1 x + \beta$ $(1):\quad$ $\displaystyle \leadsto \ \$ $\displaystyle y$ $=$ $\displaystyle -\dfrac {\alpha_1} {\alpha_2} x + \dfrac {\beta} {\alpha_2}$

Setting $x = 0$ we obtain:

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

which is the $y$-intercept.

Differentiating $(1)$ with respect to $x$ 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:

 $\displaystyle m$ $=$ $\displaystyle -\dfrac {\alpha_1} {\alpha_2}$ $\displaystyle c$ $=$ $\displaystyle \dfrac {\beta} {\alpha_2}$

$\blacksquare$