Equation of Straight Line in Plane/Slope-Intercept Form/Proof 1

Proof
Let $\LL$ be the straight line defined by the general equation:


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

We have:

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 $\LL$ and is seen to be constant.

The result follows by setting: