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