Equation of Straight Line in Plane/Two-Point Form/Parametric Form

Theorem
Let $\LL$ be a straight line embedded in a cartesian plane, given in two-point form as:
 * $\dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$

Then $\LL$ can be expressed by the parametric equations:


 * $\begin {cases} x = x_1 + t \paren {x_2 - x_1} \\ y = y_1 + t \paren {y_2 - y_1} \end {cases}$

Proof
Let $P = \tuple {x, y}$ be an arbitrary point on $\LL$.

Let $t = \dfrac {x - x_1} {x_2 - x_1} = \dfrac {y - y_1} {y_2 - y_1}$.

We then have:

The result follows.