how can i rezolve the system of equations?

Question

Solve the system of equations:

$$ \left.\begin{aligned} axy+x+y &= A \\ ayz+y+z &= B \\ azx+z+x &= C \end{aligned}\right\}~~ a,A,B,C \in (0,\infty) $$

Answer 1

Multiply each line with $a$ and then add $1$:

$$ \begin{align*} a^2xy+ax+ay +1&= aA +1\\ a^2yx+ay+az +1&= aB+1 \\ a^2zx+az+ax+1 &= aC+1 \end{align*} $$

We get $$ \begin{align*} (ax+1)(ay+1)&= aA +1\\ (ay+1)(az+1)&= aB+1 \\ (az+1)(ax+1)&= aC+1 \end{align*} $$

So $$(ax+1)^2={(ax+1)^2(ay+1)(az+1)\over (ay+1)(az+1)}= {(aA +1)(aC+1)\over aB+1}$$

and thus $$x = {1\over a}\Big(\sqrt{(aA +1)(aC+1)\over aB+1}-1\Big)$$ and similary for $y$ and $z$.