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

Theorem

Let $\LL$ be the straight line in the Cartesian plane such that:

the slope of $\LL$ is $m$

the $y$-intercept of $\LL$ is $c$

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

$y = m x + c$

such that $m$ is the slope of $\LL$ and $c$ is the $y$-intercept.

By definition, the $y$-intercept of $\LL$ is $\tuple {0, c}$.

We calculate the $x$-intercept $X = \tuple {x_0, 0}$ of $\LL$:

$\ds m$

$=$

$\ds \dfrac {c - 0} {0 - x_0}$

$\ds \leadsto \ \ $

$\ds x_0$

$=$

$\ds -\dfrac c m$

$\ds \dfrac x {-c / m} + \dfrac y c$

$=$

$\ds 1$

$\ds \leadsto \ \ $

$\ds y$

$=$

$\ds m x + c$

after algebra

$\blacksquare$

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