What is the answer to ((1-z))/((1+z))=2i ?

The answer to ((1-z))/((1+z))=2i is z=-3/5-4/5 i

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((1-z))/((1+z))=2i

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

z = - \frac{3}{5} - \frac{4}{5} i

Solution steps

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

Substitute

z = x + y i

\frac{1 - ( x + y i )}{1 + x + y i} = 2 i

Multiply both sides by

1 + x + i y

\frac{1 - ( x + y i )}{1 + x + y i} (1 + x + i y) = 2 i (1 + x + i y)

Simplify

(1 - x) - i y = - 2 y + 2 i (1 + x)

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[1 - x = - 2 y - y = 2 + 2 x]

[1 - x = - 2 y - y = 2 + 2 x] : x = - \frac{3}{5}, y = - \frac{4}{5}

Substitute back

z = x + y i

z = - \frac{3}{5} - \frac{4}{5} i

(x + i y)^{2} + i x + y - 4 = 0

∣ z ∣ + z = 3 + i

(z^{2} - 4 i z - 1) = 0

∣ 3 z + 2 i ∣ = 6

∣ 3 z + 2 i ∣ = 3

What is the answer to ((1-z))/((1+z))=2i ?
The answer to ((1-z))/((1+z))=2i is z=-3/5-4/5 i