We have the equation:

$2x + 2xi + 4y - 4yi = 8$

We can treat the real and imaginary parts separately. To do so, let's group the real terms and imaginary terms.

Step 1: Group the real and imaginary parts

Real part: $2x + 4y = 8$

Imaginary part: $2xi - 4yi = 0$ This simplifies to: 2x = 4y \quad \text{(since i is a common factor)}

Step 2: Solve the system of equations

From the imaginary part, we can solve for $x$ in terms of $y$: $x = 2y$

Now substitute $x = 2y$ into the real part equation: $2(2y) + 4y = 8$ $4y + 4y = 8$ $8y = 8$ $y = 1$

Step 3: Find $x$

Substitute $y = 1$ into $x = 2y$: $x = 2(1) = 2$

Final Answer:

$x = 2, \quad y = 1$

Would you like further details or explanations on any steps?

Here are some related questions:

1. How do you approach solving systems of linear equations?
2. What are real and imaginary parts in complex numbers?
3. How do you identify real and imaginary terms in an equation?
4. How do you solve a system of equations involving complex numbers?
5. Can complex numbers be applied in physical systems?

Tip: Always separate the real and imaginary parts when solving equations involving complex numbers.