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

Proof

 * Straight-line-double-intercept-form.png

From the General Equation of Straight Line in Plane, $\LL$ can be expressed in the form:
 * $(1): \quad \alpha_1 x + \alpha_2 y = \beta$

where $\alpha_1, \alpha_2, \beta \in \R$ are given, and not both $\alpha_1, \alpha_2$ are zero.

Substituting for the two points whose coordinates we know about:

and:

We know that $\beta \ne 0$ because none of $a, b, \alpha_1, \alpha_2$ are equal to $0$.

Hence: