Indetermined system

Question

I am trying to solve the system $$\sum_{i=1}^Nx_i=\sum_{i=1}^N\frac{1}{x_i}=3$$ for $N>2$. The main aspect that confuses me is the general $N$ and I'm not sure how to proceed.

Answer 1

By Vieta's formulas the solutions of this system are the families of roots of polynomials $$ x^n+ a_1x^{n-1}+\ldots +a_n=0, $$ where $a_1=-3$ and $a_{n-1}/a_n=-3$. Taking arbitrarily coefficients $a_2,\ldots,a_{n-1}$, you will get all solutions.

Answer 2

Note that $x+\frac 1x \ge 2$ if $x$ is positive. If you add the two equations there are no solutions for $N \gt 3$ 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.