7. (12 points) Solve the system of equations. Write the greatest value of $y$ in your answer.

$$
\left\{\begin{array}{l}
3 x^{2}-x y=1 \\
9 x y+y^{2}=22
\end{array}\right.
$$
Answer: 5.5

Solution. Multiply the first equation by 3 and add to the second

$$
\left\{\begin{array} { l } 
{ 3 x ^ { 2 } - x y = 1 , | \cdot 3 } \\
{ 9 x y + y ^ { 2 } = 2 2 }
\end{array} \Longleftrightarrow \left\{\begin{array} { l } 
{ 3 x ^ { 2 } - x y = 1 , } \\
{ 9 x ^ { 2 } + 6 x y + y ^ { 2 } = 2 5 }
\end{array} \Longleftrightarrow \left\{\begin{array}{l}
3 x^{2}-x y=1 \\
3 x+y= \pm 5
\end{array}\right.\right.\right.
$$

Expressing $y$ from the last equation and substituting into the first, we find all solutions: $(1 ; 2),(-1 ;-2),\left(-\frac{1}{6} ; 5.5\right),\left(\frac{1}{6} ;-5.5\right)$.