\[
\begin{aligned}
x+mk &\equiv y+nk (mod L) \\
x+mk - (y+nk) &= L \times z (z \in Z) \\
x+mk - y-nk - Lz &= 0 \\
k(m-n) + (x-y) - Lz &= 0 \\
-k(n-m) - Lz &= -(x-y) \\
k(n-m) + Lz &= (x-y) \\
kW + Lz &= S \\
(W = n-m&, S = x-y)
\end{aligned}
\]