# Indetermined system <munderover> &#x2211;<!-- ∑ --> i = 1

Aedan Gonzales 2022-05-12 Answered
Indetermined system
$\sum _{i=1}^{N}{x}_{i}=\sum _{i=1}^{N}\frac{1}{{x}_{i}}=3$
for $N>2$. The main aspect that confuses me is the general $N$ and I'm not sure how to proceed.
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Semaj Stark
Note that $x+\frac{1}{x}\ge 2$ if x is positive. If you add the two equations there are no solutions for $N>3$ and all ${x}_{i}=1$ is the only solution for $N=3$ unless negative values are allowed for $x$. If negative values are allowed you can't say much.
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Elle Weber
By Vieta's formulas the solutions of this system are the families of roots of polynomials
${x}^{n}+{a}_{1}{x}^{n-1}+\dots +{a}_{n}=0,$
where ${a}_{1}=-3$ and ${a}_{n-1}/{a}_{n}=-3$. Taking arbitrarily coefficients ${a}_{2},\dots ,{a}_{n-1}$, you will get all solutions.