The equation of the line passing through point $ \left(1,~3\right) $, and point $ \left(4,~5\right) $ is:
$$ 2x-3y+7=0 $$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:
$$ \begin{aligned} & \left(1,~3\right) \implies x_A = 1 ~~\text{and}~~ y_A = 3 \\[1 em] & \left(4,~5\right) \implies x_B = 4 ~~\text{and}~~ y_B = 5 \end{aligned} $$After substituting into the formula, we obtain:
$$ \begin{aligned} y - y_A~&=~\frac{y_B - y_A}{x_B - x_A}(x - x_A) \\[1 em] y - 3~&=~\frac{ 5 - 3 }{ 4 - 1 } \left( x - 1 \right) \\[1 em]y - 3 ~&=~ \frac{ 2 }{ 3 } \left( x - 1 \right) \\[1 em]y - 3 ~&=~ \frac{ 2 }{ 3 }x-\frac{ 2 }{ 3 } \\[1 em]y ~&=~ \frac{ 2 }{ 3 }x + \frac{ 7 }{ 3 } \end{aligned} $$