Question

Find the real numbers $x$ and $y$ if $(x-iy)(3+5i)$ is the conjugate of $-6-24i$.

Answer 1

Given: $(x-iy)(3+5i)=-6-24i$

$\Rightarrow (x+iy)(3-5i)=-6-24i$

$\Rightarrow 3x-5xi+3yi+5y=-6-24i$

On compairing

$3x+5y=-6$ __ (1)

$-5x+3y=-24$ __ (2)

Solve (1) and (2)

$15x+25y=-30$

$-15x+9y=-72$

______________

$16y=42$

$y=\frac{42}{16}=21/8$

$3x+\frac{21\times 5}{8}=-6\Rightarrow x=(-6-\frac{105}{8})\times \frac{1}{3}$

$=\frac{-135}{8\times 3}=\frac{-45}{8}$

$\Rightarrow x=-45/8y=21/8$