value of x+y

Question

If x+yi/1+i = 7/7+i, where x and y are real numbers, what is the value of x+y?

Answer 1

(x+yi)/(1+i) = [(x+yi)(1-i)]/[(1+i)(1-i)]

 

                  = (x-ix+iy+y)/2

 

                  = ½(x+y) + ½(y-x)i

 

7/(7+i) = [7(7-i)]/[(7+i)(7-i)]

 

           = (49-7i)/50

 

           = (49/50) -(7/50)i

 

Two complex numbers are equal only when the real parts are equal and the imaginary parts are equal.

 

So, ½(x+y) = 49/50 and   ½(y-x) = -7/50

 

 x+y = 49/25

-x+y = -7/25 

 

2y = 42/25 

 y = 21/25  and x = 49/25 - y = 28/25

So, x+y = 49/25

 

Answer 2

x + yi/1 + i = 7/7 + i

x + yi + i = 1 + i

x + yi = 1

 

x = 1

y = 0