Assume I have, for example, $x = 0.7 + 0.7i$ and the equation as below: $$\tag{1} (0.5+0.5i)y + (0.5 - 0.5i)x = 0 $$
in that case $y$ has the same magnitude of $x$. At the same time, if we swap $x$ and $y$ in equation $(1)$, we will also have the same magnitude of $x$ and $y$. By this I mean to make the equation as: $$ (0.5+0.5i)x + (0.5-0.5i)y = 0 $$ Now, if we have changed the above equation into: $$\tag{2} (0.5+0.5i)y + (0.5-0.5i)x = 0.5 $$ It means the $0$ becomes a constant such as $0.5$, What should be $x$ and $y$ in function of the constant $0.5$ in order to keep the property of first equation, which is having the same magnitude for $x$ and $y$ in both cases of keeping them as it is or swapping them?.
You previously had 0.5y + 0.5x = 0 and 0.5iy - 0.5ix = 0
Now you have (1+i)y + (1-i)x = 1 Hence x+y = 1 and iy - ix = 0
In both cases it guarantees x and y to have the same magnitude.
This is also symmetric set of equations, however if you want to keep the same magnitude, then you enforce x and y to have the same real argument equals 0.5
This is a bit intuitive since you change real argument on RHS.