# 3 variables, 2 equations

Algebra Level 5

$\large \begin{cases} { \log _{ x }{ { y }^{ 2 } } +\log _{ y }{ { x }^{ 5 } } =2n-1 } \\ { \log _{ { x }^{ 2 } }{ { y }^{ 5 } } +\log _{ { y }^{ 2 } }{ { x }^{ 3 } } =n-3 } \\ \end{cases}$

Compute the sum of all values of $$n$$ such that there exist real values of $$x$$ and $$y$$ that satisfy the system of equations above.

The sum is in the form of $$\frac{p}{q}$$ for coprime positive integers $$p,q$$. Find $$p+q$$.

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