Question:
Find the real numbers $x$ and $y$ if $(x-i y)(3+5 i)$ is the conjugate of $-6-24 i$.
Solution:
Let $z=(x-i y)(3+5 i)$
$z=3 x+5 x i-3 y i-5 y i^{2}=3 x+5 x i-3 y i+5 y=(3 x+5 y)+i(5 x-3 y)$
$\therefore \bar{z}=(3 x+5 y)-i(5 x-3 y)$
It is given that, $\bar{z}=-6-24 i$
$\therefore(3 x+5 y)-i(5 x-3 y)=-6-24 i$
Equating real and imaginary parts, we obtain
$3 x+5 y=-6$ (i)
$5 x-3 y=24$ $\ldots$ (ii)
Multiplying equation (i) by 3 and equation (ii) by 5 and then adding them, we obtain
$9 x+15 y=-18$
$\frac{25 x-15 y=120}{34 x=102}$
$\therefore x=\frac{102}{34}=3$
Putting the value of $x$ in equation (i), we obtain
$3(3)+5 y=-6$
$\Rightarrow 5 y=-6-9=-15$
$\Rightarrow y=-3$
Thus, the values of $x$ and $y$ are 3 and $-3$ respectively.