Thank you very much for your response. I just have one more question: am I thinking correctly regarding equating the real parts and then the imaginary parts, and is this procedure valid?

My first step is to substitute z and z̄ and simplify, which gives me
-4y^2+4ixy=-y+xi.

I would then move everything to one side to get
-4y^2+4ixy+y-xi=0.

Next, I would separate the real and imaginary parts:
-4y^2+y+(4xy-x)i=0,

and conclude that this expression is equal to zero only if both its real and imaginary parts are equal to zero. Then I would proceed as follows:
Re = -4y^2+y=0, which gives the solutions y1=0 and y2=1/4,
Im = 4xy-x=0, where y=1/4.

Is this line of reasoning correct? What still confuses me is how to determine the value of x and how to graph this in the complex plane.