What is the ordered pair of integers $(x,y)$ for which $12x + 21y = 15$ and $21x + 12y = 51$?
Simplify both equations by dividing by 3: \begin{align*}
4x + 7y &= 5 \\
7x + 4y &= 17.
\end{align*} We solve this system using the elimination method.  Multiply the first equation by 7 and the second equation by $-4$ to obtain \begin{align*}
28x + 49y &= 35 \\
-28x -16y &= -68.
\end{align*} Adding the equations gives $33y=-33$, so $y=-1$.  Substituting $y=-1$ into either equation and solving, we get $x=3$.  Therefore, $(x,y)=\boxed{(3,-1)}$.