For a simple, really concrete example you can also look at:
A=k[x,y,u,v]/(xy+ux2+vy2)A=k[x,y,u,v]/(xy+ux2+vy2), X=Spec(A)X=Spec(A), I=(x,y)I=(x,y), U=D(I)U=D(I).
Then the functions f=frac−vx=fracy+uxy2f=frac−vx=fracy+uxy2 and g=frac−uy=fracx+vyx2g=frac−uy=fracx+vyx2 are defined on UU. But yf+xg=1yf+xg=1, so UU is affine!
Cheers,
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