# Thread: Complex Equations with a complex number and its conjugate

1. ## Complex Equations with a complex number and its conjugate

Hi,
I'm wondering how to solve equations such as the following:
$\displaystyle 2z - \bar{z}i = 1+4i$
I tried replacing z with (x+yi) and $\displaystyle \bar{x}$ with (x-yi) but that just gave me two unknowns.
My textbook has something similar in one of the exercises but no examples for it.
Thanks

2. Originally Posted by alexgeek
Hi,
I'm wondering how to solve equations such as the following:
$\displaystyle 2z - \bar{z}i = 1+4i$
I tried replacing z with (x+yi) and $\displaystyle \bar{x}$ with (x-yi) but that just gave me two unknowns.
My textbook has something similar in one of the exercises but no examples for it.
Thanks
$\displaystyle 2(x+iy) - (x-iy)i = 1 +4i$

$\displaystyle 2x+2yi - xi - y = 1 +4i$

$\displaystyle (2x-y) +i(2y-x) = 1 +4i$

so complex numbers are equal if

$\displaystyle z_1 = x_1 + i y_1$

$\displaystyle z_2 = x_2 + i y_2$

$\displaystyle z_1 = z_2 \Rightarrow x_1 = x_2 \; \; and \; \; y_1 = y_2$

so you have ...

$\displaystyle 2x+y = 1$

$\displaystyle 2y-x = 4$

solve that (in any way that you know) and you will get your complex number Z = x+ iy

solution : $\displaystyle x= -\frac {2}{5} \;\; y= \frac {9}{5}$

3. Ahh I got that far and didn't see it like that. Sort of like equating coefficients.
Thanks

4. Originally Posted by alexgeek
Ahh I got that far and didn't see it like that. Sort of like equating coefficients.
Thanks
you're welcome