Does anyone know how can I solve for $w$ after a linear fractional mapping?

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We have $\frac{wi+1-i}{-w+2i} = \frac{-z+1}{z-1}$

We need to get this in the form of $w = \frac{az+b}{cz+d}$

This is what I tried,

$=\frac{wi+1-i}{-w+2i} = \frac{-z+1}{z-1}$

$=\frac{wi-\frac{1}2i+1}{-w} = -z+1/z-1$

$=\frac{wi-\frac{1}2i}{-w} = \frac{-z+1}{z-1}-1$

$=wi-\frac{1}2i = -w\frac{-z+1}{z-1}-1$

$=wi = -w\frac{-z+1}{z-1}-1+\frac{1}2i$

$=wi(z-1) = -w-1+\frac{1}2i$

$=wiz-wi = -w-1+\frac{1}2i$

$=w+wiz-wi = -1+\frac{1}2i$

$=w+wi-wi = -1+\frac{1}2i * z$

$=w = -1+\frac{1}2i * z$

Does anyone have any ideas of how to get to $w = \frac{az+b}{cz+d}$