To solve the system of equations:

1. $x^2 + 22xy + 3y^2 + 4x + 6y + 27 = 0$
2. $3x^2 + 10xy + 8y^2 + 2x + 9y + 7 = 0$

Step 1: Rearrange the equations

Group terms systematically:

1. $x^2 + 22xy + 3y^2 + 4x + 6y + 27 = 0$
2. $3x^2 + 10xy + 8y^2 + 2x + 9y + 7 = 0$

Step 2: Use substitution or elimination

We can solve this nonlinear system using substitution or elimination. I will attempt solving for one variable in terms of another (e.g., $y$ in terms of $x$).

Step 3: Eliminate one variable

Multiply the first equation by $3$ and the second equation by $1$ (to align $x^2$): $3(x^2 + 22xy + 3y^2 + 4x + 6y + 27) = 3x^2 + 66xy + 9y^2 + 12x + 18y + 81 = 0$ $3x^2 + 10xy + 8y^2 + 2x + 9y + 7 = 0$

Subtract the second equation from the scaled first equation: $(66xy - 10xy) + (9y^2 - 8y^2) + (12x - 2x) + (18y - 9y) + (81 - 7) = 0$ $56xy + y^2 + 10x + 9y + 74 = 0$

This is the reduced equation.

Step 4: Solve the reduced system

Now solve:

1. $56xy + y^2 + 10x + 9y + 74 = 0$
2. Substitute this into the other equation to eliminate one variable.

I will compute the exact solutions. sympy import symbols, Eq, solve

Define variables

x, y = symbols('x y')

Define equations

eq1 = Eq(x2 + 22xy + 3*y2 + 4x + 6y + 27, 0) eq2 = Eq(3x**2 + 10xy + 8y**2 + 2x + 9y + 7, 0)

Solve the system of equations

solution = solve((eq1, eq2), (x, y))

solutionThe solution to the given system of equations involves highly complex expressions with nested radicals and involves complex numbers. The exact solutions for $x$ and $y$ are too elaborate to interpret directly in this format.

To simplify further or obtain numerical approximations, I recommend solving these equations numerically.

Would you like me to provide numerical approximations or details of the intermediate steps?