Equation of Straight Line in Plane/Two-Intercept Form/Proof 2

Theorem

Let $\LL$ 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 $\LL$ can be described by the equation:

$\dfrac x a + \dfrac y b = 1$

Proof

By definition, $\LL$ passes through $\tuple {a, 0}$ and $\tuple {0, b}$.

From the Equation of Straight Line in Plane through Two Points, $\LL$ can be expressed in the form:

\(\ds \dfrac {y - 0} {x - a}\)

\(=\)

\(\ds \dfrac {b - 0} {0 - a}\)

\(\ds \leadsto \ \ \)

\(\ds -a y\)

\(=\)

\(\ds b x - b a\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac x a + \dfrac y b\)

\(=\)

\(\ds 1\)

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {III}$. Analytical Geometry: The Straight Line: Equation of a Straight Line: Intercept form