Solving the system

$\{\begin{array}{l}18x{y}^{2}+{x}^{3}=12\\ 27{x}^{2}y+54{y}^{3}=38\end{array}$

and I wonder whether there is some slick method to find the only real root $(x,y)=(2,1/3)$ without relying on Cardano's formula, ideally giving some intuition. This closely reassembles some kind of elliptic curves, so I'm tagging it as such, please remove if wrong. Number theoretic approaches are welcome.

$\{\begin{array}{l}18x{y}^{2}+{x}^{3}=12\\ 27{x}^{2}y+54{y}^{3}=38\end{array}$

and I wonder whether there is some slick method to find the only real root $(x,y)=(2,1/3)$ without relying on Cardano's formula, ideally giving some intuition. This closely reassembles some kind of elliptic curves, so I'm tagging it as such, please remove if wrong. Number theoretic approaches are welcome.