Find all complex numbers $z$ which satisfy the following equation
$(i - z) (1 + 2 i) + (1 - z i) (3 - 4 i) = 1 + 7 i$

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Solution

Rewriting the given equation, we have
$z (- 1 - 2 i - 3 i - 4) + (i - 2 + 3 - 4 i - 1 - 7 i) = 0$
or $z = \frac{10 i}{- 5 (1 + i)}$ = $\frac{- 2 i (1 - i)}{1 + 1}$ = $- 1 - i$

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