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

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

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

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

z = 1 - 2 i

Solution steps

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

Substitute

z = x + y i

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

Apply fraction cross multiply: if

\frac{a}{b} = \frac{c}{d}

then

a \cdot d = b \cdot c

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

Expand

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

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

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

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

Substitute back

z = x + y i

z = 1 - 2 i

5 x - 5 i = 4 + 7

(z + 1)^{2} = 3 + 4 i

7 x + x i - 3 y i = 3 y + 9 i

z^{2} = - i

z^{2} = - 8 i

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