# Complex Algebra

• Oct 18th 2010, 09:40 PM
Haven
Complex Algebra
I'm a little rusty in complex algebra I was wondering how i would go about solving this equation for $\displaystyle w$

$\displaystyle \frac{w+i}{w-i} = \frac{-1+i}{1-i}\left(\frac{z+1}{z-1}\right)$

The correct answer being $\displaystyle w = \frac{i-z}{i+z}$
• Oct 18th 2010, 09:56 PM
Gusbob
Multiply both sides by (w-i). Then move both terms to one side. Expand the term with (w-i) and factorise w.
• Oct 18th 2010, 10:23 PM
Haven
I'm not sure exactly what you're getting at, could you write some of the steps out?
• Oct 18th 2010, 10:27 PM
Unknown008
The long way round?

$\displaystyle \dfrac{w+i}{w-i} = \dfrac{-1+i}{1-i}\left(\dfrac{z+1}{z-1}\right)$

$\displaystyle \dfrac{w+i}{w-i} = \dfrac{-z-1+iz+i}{z-1-iz+i}$

$\displaystyle (w+i)(z-1-iz+i) = (-z-1+iz+i)(w-i)$

$\displaystyle w(z-1-iz+i) + i(z-1-iz+i) = w(-z-1+iz+i) - i(-z-1+iz+i)$

$\displaystyle w(z-1-iz+i) - w(-z-1+iz+i) = - i(-z-1+iz+i) - i(z-1-iz+i)$

$\displaystyle w(z-1-iz+i +z+1-iz-i) = - i(-z-1+iz+i + z-1-iz+i)$

$\displaystyle w(2z-2iz) = - i(-2+2i)$

$\displaystyle 2w(z-iz) = - 2i(-1+i)$

$\displaystyle w(z-iz) = i +1$

$\displaystyle w = \dfrac{i + 1}{z-iz} = \dfrac{i}{z}$

Are you sure this is the correct question?
• Oct 18th 2010, 10:39 PM
Gusbob
Thanks for taking the time to work it out, Unknown008!

Sorry I was off by -1 on the RHS. it should be $\displaystyle -\frac{1+i}{1-i}\frac{z+1}{z-1}$, but i have the general method now. thanks