Hi niaren!

Have you tried solving your equations with wolframalpha?

The exact solution of your first set equations can for instance be found like this:

{\frac{1}{z} + x = 3 \frac{c^3+1}{c^3-1}, \frac{x}{z} + y = 3, \frac{y}{z} = \frac{c^3+1}{c^3-1}} - Wolfram|Alpha

(Click on the button "Exact forms" to get the exact solution.)