Find the values of x and y if `2x+4iy = -i^(3)x-y+3`

Answer

To solve the equation \(2x + 4iy = -i^3 x - y + 3\) for the values of \(x\) and \(y\), we will follow these steps:

### Step 1: Simplify the equation
First, we need to simplify the term \(-i^3\). We know that:
\[ i^3 = -i \]
Thus, we can rewrite the equation as:
\[ 2x + 4iy = -(-i)x - y + 3 \]
This simplifies to:
\[ 2x + 4iy = ix - y + 3 \]

### Step 2: Separate real and imaginary parts
Now we can separate the real and imaginary parts of both sides of the equation. The left-hand side has:
- Real part: \(2x\)
- Imaginary part: \(4y\)

The right-hand side has:
- Real part: \(3 - y\)
- Imaginary part: \(x\)

Thus, we can equate the real parts and the imaginary parts:
1. \(2x = 3 - y\) (Equation 1)
2. \(4y = x\) (Equation 2)

### Step 3: Substitute and solve the equations
From Equation 2, we can express \(x\) in terms of \(y\):
\[ x = 4y \]
Now, substitute \(x\) into Equation 1:
\[ 2(4y) = 3 - y \]
This simplifies to:
\[ 8y = 3 - y \]
Now, add \(y\) to both sides:
\[ 8y + y = 3 \]
This gives:
\[ 9y = 3 \]
Now, divide both sides by 9:
\[ y = \frac{3}{9} = \frac{1}{3} \]

### Step 4: Find the value of \(x\)
Now that we have \(y\), we can find \(x\) using Equation 2:
\[ x = 4y = 4 \times \frac{1}{3} = \frac{4}{3} \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:
\[ x = \frac{4}{3}, \quad y = \frac{1}{3} \]