7. (10 points) Solve the system of equations. In your answer, write the greatest value of $y$.

$$
\left\{\begin{array}{l}
x^{2}+3 x y-y^{2}=27 \\
3 x^{2}-x y+y^{2}=27
\end{array}\right.
$$

Answer: 3
Solution. Subtracting the first equation from the second, we get $2 x^{2}-4 x y+2 y^{2}=0$, from which $x=y$. Substituting into the first equation, we get the answer.