How many ordered triples (x,y,z) of integers satisfy the system of equations below? 
\[\begin{array}{l} x^2-3xy+2y^2-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array}\]
$\text{(A)} \ 0 \qquad  \text{(B)} \ 1 \qquad  \text{(C)} \ 2 \qquad \\ \text{(D)}\ \text{a finite number greater than 2}\qquad\\ \text{(E)}\ \text{infinitely many}$
Sum of three equations, 
$x^2-2xy+2y^2+6yz+9z^2 = (x-y)^2+(y+3z)^2 = 175$
(x,y,z) are integers, ie. $175 = a^2 + b^2$, 
$a^2$:   1,    4,    9,   16,    25,   36,   49,   64,    81,   100,   121,   144,   169
$b^2$:  174,  171,  166,  159,  150,  139,  126,   111,   94,   75,    54,    31,    6
so there is NO solution 
[Wwei.yu](https://artofproblemsolving.com/wiki/index.php?title=User:Wwei.yu&action=edit&redlink=1) ([talk](https://artofproblemsolving.com/wiki/index.php?title=User_talk:Wwei.yu&action=edit&redlink=1)) 22:09, 28 March 2020 (EDT)Wei