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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