I have the system of these three equations:
$$ax = y+z$$ $$by = x+z$$ $$cz = x+y$$
How do I find all $a$, $b$ and $c$ for which the system has real, positive solutions for $x$, $y$ and $z$?
As a comparison, I have a simpler system:
$$ax = y$$ $$by = x$$
For this simpler system, with a substitution I get that $ab = 1$ yields a system that has real positive solutions. I'd like to find something similar for the above system as well. Basically, I'm looking for the relation between $a$, $b$ and $c$, independent from $x$, $y$ and $z$.
This is somewhat similar to this other question, only my equations are different.