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

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

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

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

z = 1 - i

Solution steps

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

Substitute

z = x + y i

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

Multiply both sides by

x - 1 + i y

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

Expand

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

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

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

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

Substitute back

z = x + y i

z = 1 - i

5 x + 6 i = - 20 + 2

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

z^{2} - (3 + 6 i) z - (4 + 6 i) = 0

25 - 49 i = x^{2} + y^{2} i

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

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