The number of pairs (x,y) where both x and y are real satisfying x2 + y2 + 2 = ( 1+x )(1 + y) is
1
1.Given x2 + y2 + 2 = 1 + x + y + xy. Mutiplying both sides by 2 and rewriting, we get
(x - y)2 + (x - 1)2 + (y - 1)2 = 0. As x, y are real, this implies x - y = 0, x - 1 = 0 and y - 1 = 0
i.e., x = y = 1 is the unique solution.