Determine real values of x and y
`(x-yi)= (2+i)/(1+i)`

Answer

To solve the equation \( (x - yi) = \frac{2 + i}{1 + i} \), we will follow these steps:

### Step 1: Simplify the Right-Hand Side
We start by simplifying the expression \( \frac{2 + i}{1 + i} \). To do this, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 1 - i \).

\[ \frac{2 + i}{1 + i} \cdot \frac{1 - i}{1 - i} = \frac{(2 + i)(1 - i)}{(1 + i)(1 - i)} \]

### Step 2: Calculate the Denominator
Now, we calculate the denominator:

\[ (1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]

### Step 3: Calculate the Numerator
Next, we calculate the numerator:

\[ (2 + i)(1 - i) = 2 \cdot 1 + 2 \cdot (-i) + i \cdot 1 + i \cdot (-i) = 2 - 2i + i - i^2 \]

Since \( i^2 = -1 \), we have:

\[ = 2 - 2i + i + 1 = 3 - i \]

### Step 4: Combine the Results
Now we can combine the results from the numerator and denominator:

\[ \frac{3 - i}{2} = \frac{3}{2} - \frac{1}{2}i \]

### Step 5: Equate to the Left-Hand Side
Now we equate this to the left-hand side \( x - yi \):

\[ x - yi = \frac{3}{2} - \frac{1}{2}i \]

### Step 6: Identify Real and Imaginary Parts
From the equation, we can identify the real and imaginary parts:

- Real part: \( x = \frac{3}{2} \)
- Imaginary part: \( -y = -\frac{1}{2} \) which gives \( y = \frac{1}{2} \)

### Final Values
Thus, the real values of \( x \) and \( y \) are:

\[ x = \frac{3}{2}, \quad y = \frac{1}{2} \]

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