What is the best way to plot the solution to a system of equations in 2D and 3D, where Mathematica does as much of the heavy lifting as possible?
I'd like to give Mathematica a system, say, x==y and y-x==0 and have Mathematica give me a 2D plot showing points (1,1) and (-1,-1). Or give Mathematica the system x^2+y^2+z^2==1 and x==y and get a 3D plot of a circle about the origin in the x=y plane.
I know how to produce the plots using ParametricPlot functions, but my hope is that Mathematica can give me a easier way to quickly visualize solutions to systems of equations more complicated than the examples above.
An even better visualization would be a manipulatable plot that toggles through the equations and their intersection.