Find $x$ and $y$ if
$(x + i y) (7 - 5 i) = 9 + 4 i$

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Solution

We have,

$(x + i y) (7 - 5 i) = 9 + 4 i$

$7 x - 5 x i + 7 y i - 5 y i^{2} = 9 + 4 i$

$7 x - (5 x - 7 y) i - 5 y (- 1) = 9 + 4 i$

$(7 x + 5 y) - (5 x - 7 y) i = 9 + 4 i$

On comparing real and imaginary part, we get

$7 x + 5 y = 9$ ……. (1)

$5 x - 7 y = - 4$ ……. (2)

On solving equation (1) and (2), we get

$x = \frac{43}{74}, y = \frac{73}{74}$

Hence, this is the answer.

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