Systems of Congruences \begin{cases} \overline{xyz138} \equiv 0 \mod7 \\ \overline{x1y3z8} \equiv 5 \mod11 \\ \overline{138xyz} \equiv 6 \mod13 \end{cases}
I worked my way up to this:
\begin{cases} 2000x+200y+20z \equiv 2 \mod7 \\ 100000x+1000y+10z \equiv 4 \mod11 \\ 2000x+200y+20z \equiv 7 \mod 13 \end{cases}
I tried to subtract one equation from the other,  but got nothing. No idea what to do.

If your workings are correct, you can rewrite your equations as:
\begin{cases}
20(\overline{xyz}) &\equiv 2 \pmod 7\\
10(x+y+z) &\equiv 4 \pmod {11}\\
20(\overline{xyz}) &\equiv 7 \pmod {13}
\end{cases}
which reduces to
\begin{cases}
\overline{xyz} &\equiv 5 \pmod 7\\
x+y+z &\equiv 7 \pmod {11}\\
\overline{xyz} &\equiv 1 \pmod {13}
\end{cases}
By CRT, $\overline{xyz} \equiv 40 \pmod {91}$, so there are not many numbers to check.
