Help me with complex equations:?

Question

This one should be fun:

(x + iy + 2 + 3i)
---------------------- = i + 2
(2x + 2iy - 3)

Solve for x and y.

I know that this is a might be a lot of tedious algebra, but if you are willing to write it out and explain the approach you took, I will gladly give you 10 points. Thanks

Answer 1

First multiply both sides by the denominator 2x + 2iy - 3 to get

x +iy +2 + 3i = (2x + 2iy - 3)*(i + 2)

x +iy +2 + 3i = 2xi - 2y - 3i + 4x + 4iy - 6

Collect terms:

-3x -3iy - 2ix + 8 +6i = 0

Separate into real and imaginary parts:

(-3x + 8) + (-3y -2x + 6)i = 0

Set the real and imaginary parts each to zero. The real part gives us the value of x

x = 8/3

Put this into the imaginary part

-3y - 16/3 + 6 = 0

-9y - 16 + 18 = 0

-9y = -2

y = 2/9