Note that the slope and the $y$-intercept of $L_1$ are $3$ and $7$ respectively.
Note also that the slope and the $y$-intercept of $L_2$ are $\dfrac{12}{4}=3$ and $\dfrac{-11}{4}$ respectively.
Since $m_{L_1} = m_{L_2}$ and the $y$-intercepts of $L_1$ and $L_2$ are not equal, then $L_1 // L_2$ and they do not intersect to each other.
Since $P$ is a moving point such that $P$ is equidistant to $L_1$ and $L_2$, then the locus of $P$ is a straight line parallel to $L_1$ and $L_2$, and lies in the mid-way between $L_1$ and $L_2$.
Hence, the slope of the locus of $P = m_{L_1} = 3$ and the $y$-intercept of the locus of $P$
$\begin{array}{cl}
= & \dfrac{1}{2} \left(7 + \dfrac{-11}{4}\right) \\
= & \dfrac{17}{8}
\end{array}$
Therefore, the equation of the locus of $P$ is
$\begin{array}{rcl}
\dfrac{y-\frac{17}{8}}{x-0} & = & 3 \\
y – \dfrac{17}{8} & = & 3x \\
8y – 17 & = & 24x \\
24x – 8y + 17 & = & 0
\end{array}$
