The equation of the line passing through point $ \left(1,~4\right) $, and point $ \left(3,~-3\right) $ is:
$$ 7x+2y-15=0 $$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:
$$ \begin{aligned} & \left(1,~4\right) \implies x_A = 1 ~~\text{and}~~ y_A = 4 \\[1 em] & \left(3,~-3\right) \implies x_B = 3 ~~\text{and}~~ y_B = -3 \end{aligned} $$After substituting into the formula, we obtain:
$$ \begin{aligned} y - y_A~&=~\frac{y_B - y_A}{x_B - x_A}(x - x_A) \\[1 em] y - 4~&=~\frac{ -3 - 4 }{ 3 - 1 } \left( x - 1 \right) \\[1 em]y - 4 ~&=~ -\frac{ 7 }{ 2 } \left( x - 1 \right) \\[1 em]y - 4 ~&=~ -\frac{ 7 }{ 2 }x + \frac{ 7 }{ 2 } \\[1 em]y ~&=~ -\frac{ 7 }{ 2 }x + \frac{ 15 }{ 2 } \end{aligned} $$