Help me solve this complex number

(x - 2i)/(2 + i) + (2 - iy)/(1 - i) = 4-2i

Pls show a step-by-step working please. Thank You!

Tackle the fractions separately:

(x - 2i)/(2 + i) = (x - 2i)(2 - i)/((2 + i)(2 - i))
= (2x - 4i - xi - 2)/5
= (2x - 2 - (4 + x)i)/5

(2 - iy)/(1 - i) = (2 - iy)(1 + i)/((1 - i)(1 + i))
= (2 + 2i - iy + y)/2
= (2 + y + (2 - y)i)/2

(x - 2i)/(2 + i) + (2 - iy)/(1 - i) = 4 - 2i

(2x - 2 - (4 + x)i)/5 + (2 + y + (2 - y)i)/2 = 4 - 2i

Multiply through by 10 to get rid of those fractions for good:

2(2x - 2 - (4 + x)i) + 5(2 + y + (2 - y)i) = 40 - 20i

4x - 4 - 2(4 + x)i + 10 + 5y + 5(2 - y)i = 40 - 20i

4x - 4 + 10 + 5y - 2(4 + x)i + 5(2 - y)i = 40 - 20i

4x + 5y + 6 + (-8 - 2x)i + (10 - 5y)i = 40 - 20i

4x + 5y + 6 + (10 - 8 - 2x - 5y)i = 40 - 20i

4x + 5y + 6 + (2 - 2x - 5y)i = 40 - 20i

Now you look at the real part:
4x + 5y + 6 = 40
4x + 5y = 34

And imaginary part:
2 - 2x - 5y = -20
2x + 5y = 22
4x + 10y = 44

And you can solve them simultaneously.

Subtract the first eqaution from the second:

4x - 4x + 10y - 5y = 44 - 34
5y = 10
y = 2

Now sub in to find x:
4x + 5*2 = 34
4x + 10 = 34
4x = 24
x = 6

So the solution is (6, 2).

Edit: mohanrao d, when multiplying a 5 through one of your brackets you only did half of it, so from there onwards your answer is incorrect.

(x - 2i)/(2 + i) + (2 - iy)/(1 - i) = 4-2i

rationalize denominators

(x - 2i)(2 - i) /5 + (2 - iy)(1 + i)/2 = 4 - 2i

multiply with 10

2(x - 2i)(2 - i) + 5(2 - iy)(1 + i) = 20(2 - i)

2(2x - 4i - xi + 2i^2) + 5(2 + 2i - iy - i^2 y = 40 - 20i

=> 2(2x - i(x + 4) - 2) + 5( 2 + y + i(2 - y)) = 40 - 20i

=> 4x - 4 + 10 + 5y + i(2 - y - x - 4) = 40 - 20i

=> 4x + 5y - i(x + y + 2) = 34 - 20i

comparing real and imaginary parts

4x + 5y = 34-----------(1)
x + y + 2 = 20 ==> x + y = 18 ==> -4x - 4y = -72 --------(2)

add (1) and (2)

y = - 38

x - 38 = 18 ==> x = 56

=> x = 56 and y = -38