KöMaL 2023 C1768

Show that the simultaneous equations $$\begin{align*} 8x^3+27y^3 & =-6\cdot 5^3, \\ \frac{3}{x}+\frac{2}{y} & =\frac{xy}{5} \end{align*}$$ have no solution if $x$, $y$ are real numbers.