Trouble with an easy complex equation

For the following equation:

$\frac{Aj}{100\sqrt{2}} -\frac{A}{100\sqrt{2}} +\frac{x}{200}-\frac{xj}{200} =0$
where $A$ is a constant and $j$ is the imaginary unit.

I thought the solution would be found by isolating the real and imaginary parts and setting them both equal to zero. I would then get the answer $\frac{2A}{\sqrt{2}}+\frac{2A}{\sqrt{2}}j$, when the answer is $\frac{2A}{\sqrt{2}}$ (This is obvious by just plugging this solution in). Why does setting the real and imaginary parts to the real and imaginary parts of the right side, which are both zero, produce the wrong answer?

Thank you.

Answer

You have: $$\frac{Aj}{100\sqrt{2}} -\frac{A}{100\sqrt{2}} +\frac{x}{200}-\frac{xj}{200} =0$$ $$\therefore\left(\frac{A}{100\sqrt{2}}-\frac{x}{200}\right)j+\left(\frac{x}{200}-\frac{A}{100\sqrt{2}}\right) =0$$ Both real and imaginary components must equal zero which both lead to the answer: $$\frac{x}{200}=\frac{A}{100\sqrt{2}}$$ Which simplifies to your desired answer.

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In general, if: $$a+bj=0$$ Then this implies that $a=0$ and $b=0$

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Even more generally, if: $$\color{red}{a}+\color{blue}{b}j=\color{red}{c}+\color{blue}{d}j$$ Then this implies that $\color{red}{a}=\color{red}{c}$ and $\color{blue}{b}=\color{blue}{d}$