How do I solve the following problem:
Find the biggest value of z:
$$ \begin{cases} \begin{align} x+\frac{1}{y}&=10 \\ y+\frac{1}{z}&=10 \\ z+\frac{1}{x}&=10 \end{align} \end{cases} $$
Answer is $5 + \sqrt{24}$.
2026-03-26 01:10:26.1774487426
Find biggest value of $z$, equation system
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Hint: by brute force:
$$ \begin{align} z = 10 - \frac{1}{x} & = 10 - \cfrac{1}{10 - \cfrac{1}{y}} = 10 - \cfrac{1}{10 - \cfrac{1}{10-\cfrac{1}{z}}} \end{align} $$
After routine manipulations, the equation reduces to $\;99z^2-990z+99=0\,$.