Question

If \[ \frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y}{3-i}=i, \] then \(x+y=\)

• A. \(\dfrac{19}{63}\)
• B. \(\dfrac{9}{13}\)
• C. \(\dfrac{49}{23}\)
• D. \(\dfrac{49}{73}\)

Hint:

In complex number equations, rationalize the denominators first and then compare real and imaginary parts separately to obtain two linear equations in \(x\) and \(y\).

Answer 1

The Correct Option is C

Step 1: Rationalize both fractions and separate real/imaginary parts.
Multiply the first term by \(\tfrac{3-i}{3-i}\) and the second by \(\tfrac{3+i}{3+i}\). After simplification, equate real parts to 0 and imaginary parts to 1 to get a system in \(x\) and \(y\).

Step 2: Solve the system.
The system yields \(x = \tfrac{46}{23}\) and \(y = \tfrac{3}{23}\), so \[x + y = \frac{46+3}{23} = \frac{49}{23}.\]
\[\boxed{x+y = \frac{49}{23}}\]