Derivation of the Parametric Form of the Equation of a Line From Its Two-Point Form?

Question

Wikipedia's documentation on the parametric form of a linear equation states in the paragraph between two different sets of equations available for use in determining a line's parametric equations that the latter form 'can also be related to the two-point form, where $T = p - h$, $U = h$, $V = q - k$, and $W = k$.' How would one derive this relation — i. e.: how would one derive the parametric form of the equation for a line from the two-point form of the equation for a line?

Answer 1

If the two point form is

$$(y-y_1)=\frac{y_2-y_1}{x_2-x_1} (x-x_1)$$

We can write it as

$$\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$$

Since they are equal, we set them both equal to $t$:

$$y-y_1=(y_2-y_1)t\Rightarrow y=y_1+(y_2-y_1)t $$ $$x-x_1=(x_2-x_1)t\Rightarrow x=x_1+(x_2-x_1)t$$

This is then the parametric form.