Question

Find the real values of x and y for which (1+i)y²+(6+i)=(2+i)x

Answer 1

The value of x and y is x =5 and y=+2 or - 2

Answer 2

Answer:  The real values of x and y are

x = 5   and   y = 2, -2.

Step-by-step explanation:  We are given to find the real values of x and y or which the following equation holds :

$(1+i)^2+(6+i)=(2+i)x~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)$

We will be using the following value :

$i^2=-1.$

From equation (i), we have

$(1+i)y^2+(6+i)=(2+i)x\\\\\Rightarrow y^2+iy^2+6+i=2x+ix\\\\\Rightarrow (y^2+6)+i(1+y^2)=2x+ix.$

Comparing the real and imaginary values from both sides of the above equation, we get

$y^2+6=2x~~~\Rightarrow y^2=2x-6~~~~~~~~~~~~~~~~~~~~~~~~(ii)\\\\1+y^2=x~~~\Rightarrow y^2=x-1~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)$

Comparing equations (ii) and (iii), we get

$2x-6=x-1\\\\\Rightarrow 2x-x=-1+6\\\\\Rightarrow x=5.$

From equation (iii), we get

$y^2=5-1=4\\\\\Rightarrow y=\pm2.$

Thus, the real values of x and y are

x = 5   and   y = 2, -2.