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:

$$ \begin{aligned} & \left(1,~1\right) \implies x_A = 1 ~~\text{and}~~ y_A = 1 \\[1 em] & \left(-4,~-3\right) \implies x_B = -4 ~~\text{and}~~ y_B = -3 \end{aligned} $$

After substituting into the formula, we obtain:

$$ \begin{aligned} y - y_A~&=~\frac{y_B - y_A}{x_B - x_A}(x - x_A) \\[1 em] y - 1~&=~\frac{ -3 - 1 }{ -4 - 1 } \left( x - 1 \right) \\[1 em]y - 1 ~&=~ \frac{ 4 }{ 5 } \left( x - 1 \right) \\[1 em]y - 1 ~&=~ \frac{ 4 }{ 5 }x-\frac{ 4 }{ 5 } \\[1 em]y ~&=~ \frac{ 4 }{ 5 }x + \frac{ 1 }{ 5 } \end{aligned} $$

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