We have the following system of equations:

$\text{(1)} \quad 3 - 2zx - 4zy = 0$ $\text{(2)} \quad 2 - 4zx - 2zy = 0$ $\text{(3)} \quad x^2 + 4xy + y^2 = 66$

Let's break these down step by step.

Step 1: Solve for $z$ in terms of $x$ and $y$ from the first two equations.

From the first equation, we can isolate $z$: $3 - 2zx - 4zy = 0$ $2zx + 4zy = 3$ $z(2x + 4y) = 3$ $z = \frac{3}{2x + 4y}$

Now, from the second equation: $2 - 4zx - 2zy = 0$ $4zx + 2zy = 2$ $z(4x + 2y) = 2$ $z = \frac{2}{4x + 2y} = \frac{1}{2x + y}$

Step 2: Set the two expressions for $z$ equal.

Since both expressions represent $z$, we can set them equal to each other: $\frac{3}{2x + 4y} = \frac{1}{2x + y}$

Step 3: Solve for $x$ and $y$.

Cross-multiply to solve: $3(2x + y) = 1(2x + 4y)$ $6x + 3y = 2x + 4y$ Simplify: $6x - 2x = 4y - 3y$ $4x = y$

So, we now know that $y = 4x$.

Step 4: Substitute $y = 4x$ into the third equation.

Substitute $y = 4x$ into $x^2 + 4xy + y^2 = 66$: $x^2 + 4x(4x) + (4x)^2 = 66$ $x^2 + 16x^2 + 16x^2 = 66$ $33x^2 = 66$ $x^2 = 2$ $x = \pm \sqrt{2}$

Step 5: Find the value of $y$.

Since $y = 4x$, we substitute $x = \pm \sqrt{2}$: $y = 4(\pm \sqrt{2}) = \pm 4\sqrt{2}$

Step 6: Find $z$.

Now, substitute $x = \pm \sqrt{2}$ and $y = \pm 4\sqrt{2}$ into one of the expressions for $z$. Using $z = \frac{3}{2x + 4y}$, we get: $z = \frac{3}{2(\pm \sqrt{2}) + 4(\pm 4\sqrt{2})}$ $z = \frac{3}{2\sqrt{2} \pm 16\sqrt{2}} = \frac{3}{(2 \pm 16)\sqrt{2}}$ Thus, $z = \frac{3}{18\sqrt{2}}$ or $z = \frac{3}{-14\sqrt{2}}$.

Simplifying: $z = \frac{1}{6\sqrt{2}} \quad \text{or} \quad z = \frac{-1}{14\sqrt{2}}$

Final Solutions:

The possible solutions for $(x, y, z)$ are:

1. $x = \sqrt{2}, y = 4\sqrt{2}, z = \frac{1}{6\sqrt{2}}$
2. $x = -\sqrt{2}, y = -4\sqrt{2}, z = \frac{-1}{14\sqrt{2}}$

Let me know if you need more clarification or a step-by-step breakdown of the solution!

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Here are some related questions for you:

1. How do you approach solving systems of nonlinear equations?
2. How do you handle substitutions and eliminations in a system like this?
3. Can you solve similar problems when the equations involve higher degrees (like cubic terms)?
4. What’s the importance of symmetry in such systems of equations?
5. How do you verify that your solutions are correct after solving?

Tip: When solving nonlinear systems like this, always check if substitution or elimination works best based on the structure of the equations.