solve an equation in complex plane

Question

$a(5-i)+b=ai-3$, $a$ and $b$ are conjugate complex number.

Find $a$ and $b$.

I have tried several methods to solve it but it stuck.

How find the relationship of a and b in the equation with the complex number?

Answer 1

$a(5-i)+b=ai-3$

so

$b=a(2i-5)-3$

but

$b=x+iy$

and

$a=x-iy$

$x+iy=(x-iy)(2i-5)-3=2ix-5x+2y+5iy-3=$

$=(2y-5x-3)+i(2x+5y)$

So you must solve the system

$x=2y-5x-3$

and

$y=2x+5y$

$x=\frac{y}{3}-\frac{1}{2}$

$y=\frac{2y}{3}-1+5y=\frac{17}{3}y-1$

To sum up

$y=\frac{3}{14}$

$x=-\frac{3}{7}$

Answer 2

If you conjugate the starting equation $a(5-i)+b=ai-3$, you get:

$$b(5+i)+a=-bi-3$$ so you have to solve this system...

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We have $b = 2ai-5a-3$ (from first equation) and put it in a second, so $$(2ai-5a-3)(5+2i)+a=-3$$

so $$-28a -15-6i =-3 \implies a = {6-3i \over 14}$$