How many ordered triples $(x,y,z)$ of integer solutions are there to the following system of equations? $$ \begin{align} x^2+y^2+z^2&=194 \\ x^2z^2+y^2z^2&=4225 \end{align} $$
2026-03-29 04:48:56.1774759736
Ordered triples solution to system of equations
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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.