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 separate the real and imaginary parts and equate them.

### Step 1: Write down the equation
We start with the given equation:
\[ x + 4yi = ix + y + 3 \]

### Step 2: Rearrange the equation
Rewrite the equation by moving all terms to one side:
\[ x + 4yi - ix - y - 3 = 0 \]

### Step 3: Separate real and imaginary parts
Now, we can separate the real and imaginary parts. The left side can be expressed as:
- Real part: \( x - y - 3 \)
- Imaginary part: \( 4y - x \)

Thus, we can set up the following equations:
1. \( x - y - 3 = 0 \) (Real part)
2. \( 4y - x = 0 \) (Imaginary part)

### Step 4: Solve the first equation for \( x \)
From the first equation:
\[ x - y - 3 = 0 \implies x = y + 3 \]

### Step 5: Substitute \( x \) in the second equation
Now substitute \( x \) from the first equation into the second equation:
\[ 4y - (y + 3) = 0 \]

### Step 6: Simplify the equation
Simplifying gives:
\[ 4y - y - 3 = 0 \implies 3y - 3 = 0 \]

### Step 7: Solve for \( y \)
Now, solve for \( y \):
\[ 3y = 3 \implies y = 1 \]

### Step 8: Substitute \( y \) back to find \( x \)
Now substitute \( y = 1 \) back into the equation for \( x \):
\[ x = y + 3 = 1 + 3 = 4 \]

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

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