How should I go around solving this system of non-linear equations? $$x+\frac{1}{y} =2 \frac{1}{3}$$ $$y+\frac{1}{z}=2\frac{3}{4}$$ $$z+\frac{1}{x}=-3\frac{1}{2}$$
I managed to solve it using substitution (i.e. eliminating $x$ and $y$ in the first equation), but it seems like there might be an easier method, consider the equations appear symmetrical
Hint: From the first equation we get $$y=\frac{1}{\frac{7}{3}-x}$$ and from the last one $$z=-\frac{7}{2}-\frac{1}{x}$$ so we get $$\frac{1}{\frac{7}{3}-x}+\frac{1}{-\frac{7}{2}-\frac{1}{x}}=\frac{11}{4}$$ Can you finish?