Find the real values of x and y for which:

Solution:

$(1-i) x+(1+i) y=1-3 i$

$\Rightarrow x-i x+y+i y=1-3 i$

$\Rightarrow(x+y)-i(x-y)=1-3 i$

Comparing the real parts, we get

$x+y=1 \ldots(i)$

Comparing the imaginary parts, we get

$x-y=-3 \ldots$ (ii)

Solving eq. (i) and (ii) to find the value of x and y

Adding eq. (i) and (ii), we get

$x+y+x-y=1+(-3)$

$\Rightarrow 2 x=1-3$

$\Rightarrow 2 x=-2$

$\Rightarrow x=-1$

Putting the value of x = -1 in eq. (i), we get

$(-1)+y=1$

$\Rightarrow y=1+1$

$\Rightarrow y=2$