System of 3 equations

Question

I am doing thermal calculation in electronics and when trying to device a general formula for equivalent system resistance to air flow of a part of real system, I ended with this system of three equations ($x,y,z$ are unknown; $a, b, c$ are positive parameters): $$ \frac{1}{\sqrt x} + \frac{1}{\sqrt{y+z}} = a\\ \frac{1}{\sqrt y} + \frac{1}{\sqrt{x+z}} = b\\ \frac{1}{\sqrt z} + \frac{1}{\sqrt{x+y}} = c $$ There is no problem in solving $x, y, z$ in Excel solver, Matlab, or other. The problem is with analytical solution that can be useful in solving more complex systems elegantly. I was trying to do some substitutions, but with no success at all. So, question arises: is it possible at all to solve above system of equations for $x, y, z$ analytically?