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November 2nd, 2017, 03:00 PM  #1 
Member Joined: Sep 2014 From: Sweden Posts: 94 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, 03:30 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,575 Thanks: 1423 
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, 05:28 PM  #3 
Senior Member Joined: Aug 2012 Posts: 2,409 Thanks: 753 
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 05:31 PM. 
November 2nd, 2017, 08:39 PM  #4 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,299 Thanks: 960 Math Focus: Wibbly wobbly timeywimey stuff.  
November 2nd, 2017, 09:37 PM  #5 
Senior Member Joined: Aug 2012 Posts: 2,409 Thanks: 753  
November 2nd, 2017, 10:05 PM  #6  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,299 Thanks: 960 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
November 2nd, 2017, 10:06 PM  #7 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,299 Thanks: 960 Math Focus: Wibbly wobbly timeywimey stuff.  
November 22nd, 2017, 05:43 PM  #8 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 673 Thanks: 88 
I haven't seen a z with a bar on top before. What does it mean?

November 22nd, 2017, 06:16 PM  #9 
Senior Member Joined: Aug 2012 Posts: 2,409 Thanks: 753  

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complex, conjugate, equations, number, solve 
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