Solve system of equations $3(x+\frac{1}{x}) = 4(y + \frac{1}{y}) = 5(z+\frac{1}{z})$, $xy+yz+zx = 1$ Find all $x,y,z>0$ such that
$$3(x+\frac{1}{x}) = 4(y + \frac{1}{y}) = 5(z+\frac{1}{z})$$ $$xy+yz+zx = 1$$
The only solution should be $x=\frac{1}{3}$, $y = \frac{1}{2}$, $z=1$.
There is a way to do it with $x = \tan \alpha$, etc., but I would like to find an even more elementary way.
So far, I have written
$$x^2 - \frac{k}{3}x + 1 = y^2 - \frac{k}{4}y + 1 = z^2 - \frac{k}{5}z + 1 = 0$$
and noticed that at least two of $x$, $y$, $z$ must be smaller than $1$. (So in the bad cases I can express uniquely $x,y,z$.) But then?
Any help appreciated!

Rewrite the second equation $xy+yz+zx = 1$ as $z=\frac{1-xy}{x+y}$ and evaluate
$$z+\frac1z= \frac{(x^2+1)(y^2+1)}{(x+y)(1-xy)}$$
Substitute above $z+\frac1z = \frac{z^2+1}z$ into the first equation
$$\frac{3(x^2+1)}x=\frac{4(y^2+1)}y = \frac{5(z^2+1)}z \tag1$$
to get
\begin{align}
&(x^2+1)\left(\frac3{5x}- \frac{y^2+1}{(x+y)(1-xy)}\right)=0\\
&(y^2+1)\left(\frac4{5y}- \frac{x^2+1}{(x+y)(1-xy)}\right)=0\\
\end{align}
which reduce to
\begin{align}
&3x^2y+8xy^2+2x-3y=0\tag2\\
&9x^2y+4xy^2-4x+y=0\tag3\\
\end{align}
Note that 3$\times$(2) -(3) and 2$\times$(3) -(2) simplify the equations to
\begin{align}
&2xy^2+x -y=0\tag4\\
&3x^2y-2x+y=0\tag5
\end{align}
and, furthermore, with 3$x\times$(4) -2$y\times$(5)
$$3x^2+xy-2y^2=(3x-2y)(x+y)=0$$
Substitute the resulting $y=\frac32 x$ and $y=-x$ into (1) to obtain the real solutions
$$(x,y,z)=\pm \left(\frac13,\frac12,1\right)$$
