Is there a systematic way to solve the equation $x+yi=-y+xi=0$ where x,y are complex numbers? Or is it simply solved by observation, and so the answer is $x=-i$ and $y=1$? Thanks in advance.
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if $x=a+ib$, $y=c+id$ then the condition turns out to be $a+ib + ic-d = -c-id+ia-b$ so $$a-d=-c-b$$ and $$b+c=a-d$$ so together it means $$(a-d)=-(a-d)$$ or $a=d$ and $b=-c$ so $x=a+ib$ and $y=-b+ia$ there are thus infinitely many such pairs
Note: $x=-iy$ has infinitely number of solutions.
For example, $y=a+ib \implies x=-(a+ib) \cdot i$