i need to find the real numbers of x and y which satisfy the equation : 2 + x - yi / 3x + yi = 1 + 2i

Question

Struggling with this question.  

Answer 1

(2+x - y i )/(3x +yi) = 1 + 2i

((2+x-iy)(3x-iy) )/(9x²+y²)=[3x²+6x-y²-i(4yx+2y)](9x²+y²)=1+2i

 

(3x²+6x-y²)/(9x²+y²)=1

-(4xy+2y)/(9x²+y2)=2

 

3x²+6x-y²=9x²+y²
-(2xy+y)=9x²+y²

 

3x²+y²-3x=0

9x²+y²+2xy+y=0

 

Solution(s) to this system of equations is

x=-1/2, y=-3i/2,

x=-1/2,y=3i/2,

x=0, y=0,

x=1/4, y=-3/4.

Since you are looking for real numbers x and y, so x=0, y=0 and  x=1/4, y=-3/4 is the solution.