Question

If \[ \frac{(2 - i)\,x + (1 + i)}{2 + i} \;+\; \frac{(1 - 2i)\,y + (1 - i)}{1 + 2i} \;=\; 1 - 2i, \quad\text{then}\quad 2x + 4y =\;? \]

• A. \(5\)
• B. \(-2\)
• C. \(1\)
• D. \(-1\)

Hint:

When dealing with complex expressions of the form \(\frac{a+bi}{c+di}\), always multiply by the conjugate of the denominator to simplify.
- Equate real and imaginary parts separately to form solvable systems in \(x\) and \(y\).

Answer 1

The Correct Option is A

Step 1: Express each complex fraction in standard form.
We start with the following fractions: \[ \frac{(2 - i)x + (1 + i)}{2 + i} \quad\text{and}\quad \frac{(1 - 2i)\,y + (1 - i)}{1 + 2i}. \] To simplify these expressions, multiply both the numerator and the denominator by the respective conjugates: \[ 2 + i \;\to\; 2 - i, \quad 1 + 2i \;\to\; 1 - 2i. \] This step helps in simplifying the real and imaginary parts effectively. Step 2: Isolate the real and imaginary parts to form linear equations.
After rationalizing the denominators and combining like terms for \(x\) and \(y\), we equate the resulting expression to \(1 - 2i\). By comparing the real and imaginary parts on both sides, we obtain two linear equations involving \(x\) and \(y\). Step 3: Solve for \(x\) and \(y\), then find \(2x + 4y\).
By solving the system (with some detailed algebraic steps), we find that: \[ 2x + 4y = 5. \] Thus, the final result is \(\boxed{5}\).