Equation of Straight Line in Plane/Two-Intercept Form

Theorem
Let $\mathcal L$ be a straight line which intercepts the $x$-axis and $y$-axis respectively at $\tuple {a, 0}$ and $\tuple {0, b}$, where $a b \ne 0$.

Then $\mathcal L$ can be described by the equation:
 * $\dfrac x a + \dfrac y a = 1$

Proof

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

From the General Equation of Straight Line in Plane, $\mathcal L$ 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:

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

Hence: