Discussion

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

Answer: D

\(x + y = -1\).

Simplify the right side by multiplying by the conjugate of \(1 + i\):

\[ \begin{aligned} \frac{7 - 7i}{1 + i} &= \frac{(7 - 7i)(1 - i)}{2} \\ &= \frac{7 - 14i + 7i^{2}}{2} = -7i \end{aligned} \]

With \(z = x + iy\) and \(\bar{z} = x - iy\):

\[ \begin{aligned} 5z - 2\bar{z} &= 3x + 7iy \\ 3x + 7iy &= -7i \end{aligned} \]

Equating real and imaginary parts gives \(x = 0\) and \(y = -1\), so \(x + y = -1\). Check: with \(z = -i\), \(5(-i) - 2(i) = -7i\), as required.

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