(In reply to
Solution by Jer)
We can get that graph analytically.
Start by substituting the identity x^3+y^3 = (x+y)^3-3xy(x+y) into the relation:
(x+y)^3-3xy(x+y)+3(xy)^2 = (xy)^3
Rearrange terms:
(x+y)^3 - (xy)^3 = 3xy(x+y) - 3(xy)^2
Factor:
(x+y-xy) * ((x+y)^2+(x+y)*(xy)+(xy)^2) = (x+y-xy) * (3xy)
Then either x+y-xy = 0 or (x+y)^2+(x+y)*(xy)+(xy)^2 = 3xy.
The first equation is the hyperbola 1/x+1/y=1.
The second equation seems to be a degenerate curve. Trying to implicitly plot it yields two points (0,0) and (-1,-1). Changing the 3 on the right side into something bigger like 3.1 forms a graph with an egg-shaped curve going around (-1,-1)
re: Solution
