Not infinite

The given equation can be rewritten as $\dfrac{x + y}{xy} = \dfrac{1}{100} \Longrightarrow xy = 100(x + y) \Longrightarrow xy - 100x - 100y = 0 \Longrightarrow (x - 100)(y - 100) = 10000$ .

Now since $10000 = 2^{4}5^{4}$ this value has $(4 + 1)(4 + 1) = 25$ positive integer divisors, and thus $2 \times 25 = 50$ integer divisors in general, (i.e., a negative divisor mirroring each positive divisor). So we can represent $10000$ as a pairwise product $ab$ in $50$ distinct ways, assigning $a = x - 100, b = y - 100$ , which in turn gives us $50$ distinct pairs $(x,y)$ . However, one of these pairs is $(a,b) = (-100,-100) \Longrightarrow (x,y) = (0,0)$ , which would not satisfy the original equation. Thus we are left with a final total of $\boxed{49}$ ordered integer pair solutions to the given equation.