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

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

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

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

z = 2 + i

Solution steps

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

Substitute

z = x + y i

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

Apply fraction cross multiply: if

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

then

a \cdot d = b \cdot c

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

Expand

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

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

Substitute back

z = x + y i

z = 2 + i

x + i y = 3 i - 4

(11 - 3 x) + (7 + 4 y) i = - 17 + i

a + bi + \frac{1}{a + bi} = 0

i r = v

(2 + 3 i) \cdot (x + i y) = (2 + i)^{2}

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