What is the answer to (5-2i)/z =2+i ?

The answer to (5-2i)/z =2+i is z= 8/5-9/5 i

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(5-2i)/z =2+i

\frac{5 - 2 i}{z} = 2 + i

z = \frac{8}{5} - \frac{9}{5} i

Solution steps

\frac{5 - 2 i}{z} = 2 + i

Substitute

z = x + y i

\frac{5 - 2 i}{x + y i} = 2 + i

Multiply both sides by

x + y i

\frac{5 - 2 i}{x + y i} (x + y i) = 2 (x + y i) + i (x + y i)

Expand

5 - 2 i = (2 x - y) + i (x + 2 y)

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

[5 = 2 x - y - 2 = x + 2 y]

[5 = 2 x - y - 2 = x + 2 y] : x = \frac{8}{5}, y = - \frac{9}{5}

Substitute back

z = x + y i

z = \frac{8}{5} - \frac{9}{5} i

(x - y i)^{2} = 4 (x + y i)

x^{2} + i x + 2 = 0

3 (x + 5 i) = 7 y i - 6 + 8

x - 5 i + 2 = 4 - 3 i + y i

(x + y i) + (4 + 9 i) = 9 - 4 i

What is the answer to (5-2i)/z =2+i ?
The answer to (5-2i)/z =2+i is z= 8/5-9/5 i