Question

If the complex number $z = x + iy$ satisfies the equation $5z - 2\bar{z} = \frac{7 - 7i}{1 + i}$, then the value of $x + y$ is equal to

• A. 7
• B. -3
• C. 3
• D. -1
• E. -7

Hint:

Remember that \( \frac{1-i}{1+i} = -i \). This identity allows you to simplify the RHS almost instantly: \( 7 \cdot (\frac{1-i}{1+i}) = -7i \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We equate the real and imaginary parts of both sides of a complex equation.
First, simplify the fractional term on the right side.
Step 2: Key Formula or Approach:
Substitute \(z = x + iy\) and \(\bar{z} = x - iy\).
Rationalize the denominator by multiplying by its conjugate \(1 - i\).
Step 3: Detailed Explanation:
Simplify the right side (RHS):
\[ \text{RHS} = \frac{7 - 7i}{1 + i} \cdot \frac{1 - i}{1 - i} = \frac{7(1 - i)^2}{1^2 - i^2} \] Since \((1 - i)^2 = 1 - 2i + i^2 = -2i\):
\[ \text{RHS} = \frac{7(-2i)}{2} = -7i \] Simplify the left side (LHS):
\[ \text{LHS} = 5(x + iy) - 2(x - iy) = 5x + 5iy - 2x + 2iy = 3x + 7iy \] Equate LHS and RHS:
\[ 3x + 7iy = 0 - 7i \] Equating real parts: \(3x = 0 \implies x = 0\).
Equating imaginary parts: \(7y = -7 \implies y = -1\).
We need \(x + y\):
\[ x + y = 0 + (-1) = -1 \] Step 4: Final Answer:
The value of \(x + y\) is -1.