What is the answer to solvefor z, z/(1-z)=1-5i ?

The answer to solvefor z, z/(1-z)=1-5i is z= 27/29-5/29 i

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solvefor z, z/(1-z)=1-5i

solve for

z, \frac{z}{1 - z} = 1 - 5 i

z = \frac{27}{29} - \frac{5}{29} i

Solution steps

\frac{z}{1 - z} = 1 - 5 i

Substitute

z = x + y i

\frac{x + y i}{1 - ( x + y i )} = 1 - 5 i

Multiply both sides by

1 - x - i y

\frac{x + y i}{1 - ( x + y i )} (1 - x - i y) = 1 \cdot (1 - x - i y) - 5 i (1 - x - i y)

Expand

x + i y = (1 - x - 5 y) + i (- 5 - y + 5 x)

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

[x = 1 - x - 5 y y = - 5 - y + 5 x]

[x = 1 - x - 5 y y = - 5 - y + 5 x] : x = \frac{27}{29}, y = - \frac{5}{29}

Substitute back

z = x + y i

z = \frac{27}{29} - \frac{5}{29} i

2 - i x - y = 0

x^{2} - (2 - i) x + (3 - i) = 0

(x + y) - (3 x + 2 y) i = 5 + 2 i

z^{2} - (2 - i) z + (3 - i) = 0

(2 x - 5 i y) (1 + i)^{2} = 10 (2 + i)

What is the answer to solvefor z, z/(1-z)=1-5i ?
The answer to solvefor z, z/(1-z)=1-5i is z= 27/29-5/29 i