Solve the non-linear system of equations For real $x,y,z>0$ solve the system of equation
\begin{cases}
 \dfrac{1}{x}-3 y+4 z=5,\\
   \dfrac{1}{y}-4 z+5 x=3,\\
      \dfrac{1}{z}-5 x+3 y=4,
\end{cases}
It is easy to check out that
$$
x =\frac 1 5, y= \frac 1 3,  z=\frac 1 4
$$ 
is a solution. 
How I could  prove that there is no any other real solutions?
Edit.  By adding all equations we get 
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=12.
$$

eliminating $y,z$ we get for $x$:
$$125 x^4-60 x^3-58 x^2+18 x=1$$
