Multiply the equation by xy and factor:
x2y2 - x2y - xy2 + x + y - 1 = 0;
x2y(y - 1) - x(y2 - 1) + y - 1 = 0;
(y - 1)[x2y - x(y + 1) + 1] = 0;
(y - 1)[xy(x - 1) - x + 1] = 0;
(y - 1)(x - 1)(xy - 1) = 0.
Therefore, the pairs that solve the equation are
(1,t), (t,1), and (t,1/t) for any nonzero real number t.
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