We will use the formula for slope,
\begin{equation*}
m = \frac{y_2 - y_1}{x_2 - x_1}
\end{equation*}
We substitute \(\dfrac{-3}{4}\) for the slope, \(m\text{,}\) and \((1, -4)\) for \((x_1, y_1)\text{.}\) For the second point, \((x_2, y_2)\text{,}\) we use the variable point \((x, y)\text{.}\) Substituting these values into the slope formula gives us
\begin{equation*}
\frac{-3}{4}= \frac{y - (-4)}{x - 1}=\frac{y + 4}{x - 1}
\end{equation*}
To solve for \(y\) we first multiply both sides by \(x - 1\text{.}\)
\begin{equation*}
\begin{aligned}[t]
\alert{(x-1)}\frac{-3}{4}\amp
=\frac{y +4}{x - 1}\alert{(x-1)} \amp\amp\\
\frac{-3}{4}(x-1)\amp=y+4 \amp\amp\blert{\text{Apply the distributive law.}}\\
\frac{-3}{4}x+\frac{3}{4}\amp=y+4 \amp\amp \blert{\text{Subtract 4 from both sides.}}\\
\frac{-3}{4}x-\frac{13}{4}\amp=y \amp\amp \blert{\frac{3}{4}-4=\frac{3}{4}-\frac{16}{4}=\frac{-13}{4}}\\
\end{aligned}
\end{equation*}
The equation of the line is \(y=\dfrac{-13}{4}-\dfrac{3}{4}x\)