algebra precalculus - The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong?

Let's share the fun:

$(\sqrt{x^2 +y^2}) i +2x +2iy =1$

leading to the system of : $$ 2y+ \sqrt{x^2 +y^2}= 0 $$ $$2x=1$$

upon solving we get $(0.5, \sqrt{1/12})$, or $(0.5, -\sqrt{1/12})$ as possible real coordinates of the complex number which may solve said equation...