If `3x + y i= x + 2y`, then `2x - y = `

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \(3x + yi = x + 2y\) and find the value of \(2x - y\), we can follow these steps:

### Step 1: Separate Real and Imaginary Parts
The equation given is:
\[ 3x + yi = x + 2y \]
Here, we can identify the real and imaginary parts. The left-hand side has a real part \(3x\) and an imaginary part \(y\). The right-hand side has a real part \(x\) and an imaginary part \(2y\).

Since the imaginary part on the right side is not explicitly stated, we can assume it is \(0i\). Thus, we can rewrite the equation as:
\[ 3x + yi = x + 2y + 0i \]

### Step 2: Set Up the Equations
Now, we can equate the real parts and the imaginary parts:
1. Real part: \(3x = x + 2y\)
2. Imaginary part: \(y = 0\)

### Step 3: Solve for \(y\)
From the imaginary part equation:
\[ y = 0 \]

### Step 4: Substitute \(y\) into the Real Part Equation
Now substitute \(y = 0\) into the real part equation:
\[ 3x = x + 2(0) \]
This simplifies to:
\[ 3x = x \]

### Step 5: Solve for \(x\)
Subtract \(x\) from both sides:
\[ 3x - x = 0 \implies 2x = 0 \]
Thus,
\[ x = 0 \]

### Step 6: Find \(2x - y\)
Now that we have \(x = 0\) and \(y = 0\), we can find \(2x - y\):
\[ 2x - y = 2(0) - 0 = 0 \]

### Conclusion
Therefore, the value of \(2x - y\) is:
\[ \boxed{0} \]