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Algebra Difficulty 3.0 AMC 10/12 Find the answer Canada

Suppose that $x$ and $y$ are positive numbers with $xy = \frac{1}{9}$ , $x(y+1) = \frac{7}{9}$ , $y(x+1) = \frac{5}{18}$ . What is the value of $(x+1)(y+1)$ ?

(A) $\frac{11}{6}$ (B) $\frac{8}{9}$ (C) $\frac{16}{9}$ (D) $\frac{10}{9}$ (E) $\frac{35}{18}$

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.