# 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

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:

 $\, \displaystyle x = a, y = 0: \,$ $\displaystyle \alpha_1 \times a + \alpha_2 \times 0$ $=$ $\displaystyle \beta$ $\displaystyle \leadsto \ \$ $\displaystyle \alpha_1$ $=$ $\displaystyle \dfrac \beta a$
 $\, \displaystyle x = 0, y = b: \,$ $\displaystyle \alpha_1 \times 0 + \alpha_2 \times b$ $=$ $\displaystyle \beta$ $\displaystyle \leadsto \ \$ $\displaystyle \alpha_2$ $=$ $\displaystyle \dfrac \beta b$

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

Hence:

 $\displaystyle \dfrac \beta a x + \dfrac \beta b y$ $=$ $\displaystyle \beta$ substituting for $\alpha_1$ and $\alpha_2$ in $(1)$ $\displaystyle \leadsto \ \$ $\displaystyle \dfrac x a + \dfrac y a$ $=$ $\displaystyle 1$ dividing both sides by $\beta$

$\blacksquare$

## Also known as

This form of the Equation of Straight Line in Plane is also known as the intercept form.