Find x + iy: 4 + 2i + ( 1 – i)/ ( x + iy) – (9 + 7i)(3 + 2i) = (32 + i)

To find x + iy: 4 + 2i + ( 1 – i)/ ( x + iy) – (9 + 7i)(3
+ 2i) = 32 + i.

We rewrite the given expression
as:

( 1 – i)/ ( x + iy) = (9 + 7i)(3 + 2i) +
32+i-4-2i

(1-i)/(x+iy) = (9+7i)(3+2i)+ 28 -
i.

(1-i)/(x-yi) =
(27+18i+21i+14i^2)+28-i

(1-i)/(x+yi) = (13 +39i)+28-i, as
i^2= -1.

(1-i)/(x+yi) =
41+38i

(1-i)(x-yi)/(x+y^2-y^2*i^2). =
41+38i

(x-y)/(x^2+y^2) - (x+y)i/(x^2+y^2) =
41+38i(1)

We equate real parts on  both sides of (1) and
the equate imaginary parts on both sides separately:

Real
parts:  (x-y)/(x^2+y^2) = 41....(2)

Imaginary parts:
(x+y)/(x^2+y^2) = -38....(3)

(2)/(3): (x-y)/(x+y) =
41/-38

 38(x-y) =
-41(x+y).

(38+41)x =
(-41+38)y

79x = -3y.

x  =
-(3/79)y

Substitute x= -3y/79 in (x+y)/(x^2+y^2) =
-38:

(-3/79+1)y/{(-3/79)^2 +1}y^2 =
-38.

79(76)y/{3^2+79^2)y^2 =
-38.

76*79y  =
-38(3^2+79^2)y^2

y = 76*79/(-38)(6250) =
-79/3125.

Therefore x = -3y/79 = -3*-79/79*3125 =
3/3125.

Therefore x = 3/3125 and y =
-79/3125.

So x+yi =
(3/3125)+(-79/3125)i.