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.

Find all pairs of real numbers $(x,y)$ satisfying the system of equations
\begin{align*}
 \frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\
   \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4).
\end{align*}