1980 AHSME Problem 29

How many ordered triples (x,y,z) of integers satisfy the system of equations below?

\begin{array}{l} x^2-3xy+2y^2-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array}

\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)}\ \text{a finite number greater than 2}\qquad \text{(E)}\ \text{infinitely many}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.