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:
To solve this equation, we must express both sides in the standard Cartesian form \( a + bi \). For the left side, we substitute \( z = x + iy \). For the right side, we simplify the fraction by rationalizing the denominator.
Key Formula or Approach:
For \( z = x + iy \), the conjugate is \( \bar{z} = x - iy \).
Rationalize the fraction \( \frac{A}{1+i} \) by multiplying the numerator and denominator by \( (1-i) \).

Step 2: Detailed Explanation:
Simplify the right-hand side (RHS):
\[ \text{RHS} = \frac{7 - 7i}{1 + i} = \frac{7(1-i)}{1+i} \cdot \frac{1-i}{1-i} \]
\[ \text{RHS} = \frac{7(1 - 2i + i^2)}{1^2 - i^2} = \frac{7(1 - 2i - 1)}{1 + 1} = \frac{7(-2i)}{2} = -7i \]
Now, simplify the left-hand side (LHS) by substituting \( z = x + iy \):
\[ \text{LHS} = 5(x + iy) - 2(x - iy) \]
\[ \text{LHS} = 5x + 5iy - 2x + 2iy = 3x + 7iy \]
Equating LHS and RHS:
\[ 3x + 7iy = 0 - 7i \]
Comparing real and imaginary parts:
Real part: \( 3x = 0 \implies x = 0 \)
Imaginary part: \( 7y = -7 \implies y = -1 \)
Calculating \( x + y \):
\[ x + y = 0 + (-1) = -1 \]

Step 3: Final Answer:
The value of $x + y$ is -1.