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Problem 313

Algebra Difficulty 2.2 Find the answer CEMC Cayley

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

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

If we multiply the second and third equations together, we obtain $x(y+1)y(y+1)=\frac{7}{9} \cdot \frac{5}{18}$ or $xy(x+1)(y+1)=\frac{35}{162}$ . From the first equation, $xy=\frac{1}{9}$ . Therefore, $\frac{1}{9}(x+1)(y+1)=\frac{35}{162}$ or $(x+1)(y+1)=9\left(\frac{35}{162}\right)=\frac{35}{18}$ .

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