How many solutions do these equations have x y plusplus 1?

It seems there may be a typographical error or missing context in your question. If you meant to ask about the equation (xy + 1 = 0), then it has infinitely many solutions for (x) and (y) as any pair ((x, y)) that satisfies (y = -\frac{1}{x}) (for (x \neq 0)) will be a solution. If you meant a different equation or context, please provide more details for a more accurate answer.