(a) Find x and y such that 4xi + (1 + i)y = 2x + 2yi.

We are given the equation:

\[
4xi + (1 + i)y = 2x + 2yi
\]

We want to find the values of \(x\) and \(y\) that satisfy this equation.

### Step 1: Break into real and imaginary parts

Since we are dealing with complex numbers, let's equate the real and imaginary parts on both sides of the equation.

First, expand both sides of the equation:

1. The left-hand side:
\[
4xi + (1 + i)y = 4xi + y + yi
\]
This expands to:
\[
(0 + 4x)i + (y + yi)
\]

2. The right-hand side:
\[
2x + 2yi = 2x + (0 + 2y)i
\]

Now, equate the real parts and imaginary parts:

### Real Parts:
From the left-hand side, the real part is \( y \).
From the right-hand side, the real part is \( 2x \).

Thus, the equation for the real part is:
\[
y = 2x \quad \text{(Equation 1)}
\]

### Imaginary Parts:
From the left-hand side, the imaginary part is \( 4x + y \).
From the right-hand side, the imaginary part is \( 2y \).

Thus, the equation for the imaginary part is:
\[
4x + y = 2y \quad \text{(Equation 2)}
\]

### Step 2: Solve the system of equations

We now have the following system of equations:
1. \( y = 2x \)
2. \( 4x + y = 2y \)

Substitute \( y = 2x \) from Equation 1 into Equation 2:
\[
4x + 2x = 2(2x)
\]
Simplifying this:
\[
6x = 4x
\]

Subtract \( 4x \) from both sides:
\[
2x = 0
\]

Thus, \( x = 0 \).

### Step 3: Find \( y \)

Now substitute \( x = 0 \) back into Equation 1:
\[
y = 2(0) = 0
\]

### Final Answer:

The values of \( x \) and \( y \) that satisfy the equation are:
\[
x = 0, \quad y = 0
\]

---

Would you like more details or have any questions about this solution?

Here are some related questions to consider:

1. What are real and imaginary parts in complex numbers?
2. How do we typically solve systems of linear equations?
3. What are other ways to represent complex numbers (like polar form)?
4. Can we have non-zero solutions for complex systems in different setups?
5. What happens when we change coefficients in such complex equations?

**Tip:** When solving complex equations, always break them into real and imaginary parts for easier manipulation!

Find x and y such that 4xi + (1 + i)y = 2x + 2yi.

Explore how to solve the complex equation 4xi + (1 + i)y = 2x + 2yi by breaking it into real and imaginary parts. This step-by-step guide shows how to find x and y through solving systems of linear equations. Suitable for high school students in Grades 10-12.

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