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

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

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

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

z = 2 - i

Solution steps

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

Substitute

z = x + y i

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

Multiply both sides by

x - 1 + i y

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

Expand

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

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

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

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

Substitute back

z = x + y i

z = 2 - i

z^{2} - z - 1 - 3 i = 0

\frac{3 - a + 2 b}{5} - 3 i + a + bi = 4 - i

(7 - 7 i) - 8 = x - (2 + y i)

i (x) = \frac{3}{x + 3}