Ordered triples solution to system of equations How many ordered triples  $(x,y,z)$ of integer solutions are there to the following system of equations? 
$$
\begin{align}
x^2+y^2+z^2&=194 \\ 
x^2z^2+y^2z^2&=4225
\end{align}
$$

We have $x^2 + y^2 = \dfrac{4225}{z^2}$. Plugging this into the first equation, we get that
$$z^2 + \dfrac{4225}{z^2} = 194 \implies z^2 + \left(\dfrac{65}z\right)^2 = 194$$  Once you have this you should be able to proceed and get the answer.
