What is the answer to x+yi-1/(x+yi)=5+5i ?

The answer to x+yi-1/(x+yi)=5+5i is

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x+yi-1/(x+yi)=5+5i

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

(x = - 0.10191 \dots, x = 5.10191 \dots, y = 0.09792 \dots y = 4.90207 \dots)

Solution steps

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

Multiply both sides by

x + y i

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

Expand

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

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

[x^{2} - y^{2} - 1 = 5 x - 5 y 2 x y = 5 x + 5 y]

[x^{2} - y^{2} - 1 = 5 x - 5 y 2 x y = 5 x + 5 y] : (x = - 0.10191 \dots, x = 5.10191 \dots, y = 0.09792 \dots y = 4.90207 \dots)

(x = - 0.10191 \dots, x = 5.10191 \dots, y = 0.09792 \dots y = 4.90207 \dots)

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

(a + i) (b - 3 i) = 7 - 11 i

2 (x + y i) = i (3 - 4 i)

(1 - i) z + w = 3 + 2 i

z^{2} - (3 - 4 i) z - (1 + 7 i) = 0