How to determine where these two parabolas intersect

How do I find where the two parabolas intersect?

y = x^2
x = y^2

You need to find the set of points (X,Y) for which both equations are true.
How many points should you expect?? Two parabolas may have zero, one, two, four, or an infinite number of points of intersection, although one could say they are not distinct parabolas in the last case.

You can do this by substituting one equation into the other.
I will substitute the former into the latter
y = x^2 = (y^2)^2 = y^4.
y^4 - y = 0
y(y^3-1) = 0
y = 0 or y^3 - 1 = 0
y= 0 or y^3 = 1
y= 0 or y = 1
Solving for associated x values by substitution, the solution points are
(0,0) and (1,1)