Find the real numbers $x$ and $y$ if $(x-i y)(3+5 i)$ is the conjugate of $-6-24 i$
Find the real numbers $x$ and $y$ if $(x-i y)(3+5 i)$ is the conjugate of $-6-24 i$
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
$3 x+5 y=-6$.....$(i)$
$5 x-3 y=24$....$(ii)$
Multiplying equation $(i)$ by $3$ and equation $(ii)$ by $5$ and then adding them, we obtain
$9 x+15 y=-18$
${25 x-15 y=120}$
$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$
${34 x=102}$
$\therefore x=\frac{102}{34}=3$
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.
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