By adding and subtracting the two given equations, we obtain the equivalent pair of equations
$$\begin{align*} 2/x &= x^4 + 10x^2y^2 + 5y^4 \\ 1/y &= 5x^4 + 10x^2y^2 + y^4. \end{align*}$$
Multiplying the former by $x$ and the latter by $y$, then adding and subtracting the two resulting equations, we obtain another pair of equations equivalent to the given ones,
$$3 = (x+y)^5, \qquad 1 = (x-y)^5.$$
It follows that $x = (3^{1/5}+1)/2$ and $y = (3^{1/5}-1)/2$ is the unique solution satisfying the given equations.