Find the values of x and y , if `x + 4yi = ix + y + 3` where ` x ,y in R`

Answer

To solve the equation \( x + 4yi = ix + y + 3 \) where \( x, y \in \mathbb{R} \), we will compare the real and imaginary parts of both sides of the equation.

### Step 1: Rewrite the equation
Start with the given equation:
\[ x + 4yi = ix + y + 3 \]

### Step 2: Rearrange the equation
Rearranging the equation, we have:
\[ x + 4yi = 0 + (y + 3) + ix \]

### Step 3: Identify real and imaginary parts
From the left side, the real part is \( x \) and the imaginary part is \( 4y \).
From the right side, the real part is \( y + 3 \) and the imaginary part is \( x \).

### Step 4: Set up equations from real and imaginary parts
From the real parts, we have:
\[ x = y + 3 \quad \text{(1)} \]
From the imaginary parts, we have:
\[ 4y = x \quad \text{(2)} \]

### Step 5: Substitute equation (1) into equation (2)
Substituting \( x \) from equation (1) into equation (2):
\[ 4y = y + 3 \]

### Step 6: Solve for \( y \)
Rearranging gives:
\[ 4y - y = 3 \]
\[ 3y = 3 \]
\[ y = 1 \]

### Step 7: Substitute \( y \) back to find \( x \)
Now substitute \( y = 1 \) back into equation (1) to find \( x \):
\[ x = 1 + 3 \]
\[ x = 4 \]

### Final Values
Thus, the values of \( x \) and \( y \) are:
\[ x = 4, \quad y = 1 \]