Using determinants, what is the equation of the line through $(2, 3)$ and $(4, 7)$?

Explanation

1. 01

   A point $(x, y)$ is on the line exactly when it is collinear with the two points: $\begin{vmatrix} x & y & 1 \\ 2 & 3 & 1 \\ 4 & 7 & 1 \end{vmatrix} = 0$.

2. 02

   Expanding: $x(3 - 7) - y(2 - 4) + (14 - 12) = -4x + 2y + 2 = 0$, that is $y = 2x - 1$.