Originally Posted by

**HallsofIvy** First, let me point out that "equations" are not geometric objects so it makes no sense to talk about "the locus of points equidistant from two equations". You mean, as bjhopper said, the locus of point equidistant from the two lines defined by those equations.

An "angle bisector" is the line or ray that divides an angle into two equal parts. Actually when two lines intersect, they divide the plane into **four** parts so there are four angles. The locus you seek is the pair of lines that divide those four angles into equal parts. The lines defined by y= x and y= -x are at right angles to one another so the lines that bisect them are at 45 degrees to them. Those are, again as bjhopper said, the x and y axes.