I'm asked to solve the following congruence system:
$$ \begin{split} x &\equiv 2 \pmod{5}\\ 2x &\equiv 1 \pmod{7}\\ 3x + y &\equiv 4 \pmod{11} \end{split} $$
But I think that by the chinese remainder theorem, since $5,7$ and $11$ are prime, for any given $y$ the system will have a solution mod $385$. Doesn't this mean that the system will have 11 different solutions (mod $11$ for $y$ and mod $385$ for $x$)?