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November 2nd, 2017, 04:00 PM  #1 
Member Joined: Sep 2014 From: Sweden Posts: 71 Thanks: 0  Solve complex number equations with conjugate
I have this equation $\displaystyle \displaystyle z + 2\bar{z} = 2  i$, but don't really know how to solve it. My first thought was to set $\displaystyle \displaystyle z = \bar{z}$ and solve for $\displaystyle z$. And when z was figured out I put in z and $\displaystyle \displaystyle \bar{z}$, but that failed. So how do you solve these types of equations? 
November 2nd, 2017, 04:30 PM  #2 
Senior Member Joined: Sep 2015 From: Southern California, USA Posts: 1,604 Thanks: 817 
let $z=x+i y,~x,y \in \mathbb{R}$ $x + i y + 2x  2i y = 2  i$ $3x  i y = 2  i$ $3x = 2$ $y = 1$ $x=\dfrac 2 3$ $y = 1$ $z = \dfrac 2 3 + i$ 
November 2nd, 2017, 06:28 PM  #3 
Senior Member Joined: Aug 2012 Posts: 1,638 Thanks: 415 
Here's another way using just using the standard operations on the complex numbers: conjugation, Real part $\Re$ and Imaginary part $\Im$. (1) $z + 2 \bar z = 2− i$ $\ \ \ \ $ [Given] (2) $\bar z + 2 z = 2 + i$ $\ \ \ \ $ [Conjugating both sides] $3 z + 3 \bar z = 4$ $\ \ \ \ $ [Adding (1) to (2)] $z + \bar z = \frac{4}{3}$ $2 \Re (z) = \frac{4}{3}$ $\ \ \ \ $ [Using $z + \bar z = 2 \Re (z)$] $\Re (z) = \frac{2}{3}$ Now subtracting (2)  (1) gives: $z  \bar z = 2$ and then $\Im (z) = 1$ $\ \ \ \ $ [Using $z  \bar z = 2 \Im (z)$] So $z = \frac{2}{3} + i$ Last edited by Maschke; November 2nd, 2017 at 06:31 PM. 
November 2nd, 2017, 09:39 PM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,659 Thanks: 652 Math Focus: Wibbly wobbly timeywimey stuff.  
November 2nd, 2017, 10:37 PM  #5 
Senior Member Joined: Aug 2012 Posts: 1,638 Thanks: 415  
November 2nd, 2017, 11:05 PM  #6  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,659 Thanks: 652 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
November 2nd, 2017, 11:06 PM  #7 
Math Team Joined: May 2013 From: The Astral plane Posts: 1,659 Thanks: 652 Math Focus: Wibbly wobbly timeywimey stuff.  

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complex, conjugate, equations, number, solve 
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