Two points in plane/(2,3), (5,-7)/Equation for line/Exercise/Solution

The line $G$ is given (in point-vector-form) as

{(\begin{matrix} 2 \\ 3 \end{matrix}) + t ((\begin{matrix} 5 \\ - 7 \end{matrix}) - (\begin{matrix} 2 \\ 3 \end{matrix})) ∣ t \in ℝ} = {(\begin{matrix} 2 \\ 3 \end{matrix}) + t (\begin{matrix} 3 \\ - 10 \end{matrix}) ∣ t \in ℝ} .

The description of the line with an equation is hence of the form

G = {(\begin{matrix} x \\ y \end{matrix}) ∣ 10 x + 3 y = c},

where we have to determine $c$ . Plugging in the point $(\begin{matrix} 2 \\ 3 \end{matrix})$ yields $c = 29$ , therefore

G = {(\begin{matrix} x \\ y \end{matrix}) ∣ 10 x + 3 y = 29} .