The equation of the line passing through point $ \left(2,~3\right) $, and point $ \left(5,~-1\right) $ is:
$$ 4x+3y-17=0 $$To find equation of the line passing through points $ A(x_A,y_A) $ and $ B(x_B,y_B) $, we use formula:
$$ y - y_A~=~\frac{y_B - y_A}{x_B - x_A}(x-x_A) $$In this example we have:
$$ \begin{aligned} & \left(2,~3\right) \implies x_A = 2 ~~\text{and}~~ y_A = 3 \\[1 em] & \left(5,~-1\right) \implies x_B = 5 ~~\text{and}~~ y_B = -1 \end{aligned} $$After substituting into the formula, we obtain:
$$ \begin{aligned} y - y_A~&=~\frac{y_B - y_A}{x_B - x_A}(x - x_A) \\[1 em] y - 3~&=~\frac{ -1 - 3 }{ 5 - 2 } \left( x - 2 \right) \\[1 em]y - 3 ~&=~ -\frac{ 4 }{ 3 } \left( x - 2 \right) \\[1 em]y - 3 ~&=~ -\frac{ 4 }{ 3 }x + \frac{ 8 }{ 3 } \\[1 em]y ~&=~ -\frac{ 4 }{ 3 }x + \frac{ 17 }{ 3 } \end{aligned} $$