When working in the complex domain (##z = x + iy##), how does one write the equation of a line?(adsbygoogle = window.adsbygoogle || []).push({});

I have attached a problem I was working on (and have the solution), but am curious as to why the definition of a line is given by ##ax + by = c##. Are not ##x## and ##y## also variables that take on strictly real values? Should not this equation for this function (upon which an arbitrary point ##z^* = x^* + iy^*## will be reflected) be written: ## y = -i(\frac {a}{b})x + \frac {c}{b} ## since we are discussing the ##z##-plane where the imaginary axis corresponds to the value of ##y##?

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# I Equations for functions in the complex domain

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