Trying to find $x \equiv_{17} -4$, $x \equiv_{23} 3$.
OK, so $x = -4 + 17k$ for some $k$.
$-4 + 17k \equiv_{23} 3$. Since $19$ is the inverse of $17 \pmod {23}$, $k \equiv_{23} (3+4)19 \equiv 133$.
Plugging those in: $x = 13 + 17(133) = 2274$.
Now, $2274 \equiv_{17} -4$, but for $\bmod 23$, I have to make it negative $-2274 \equiv_{23} 3$, otherwise it's $20$.
What did I do wrong?