Maxing out

Algebra Level 2

\[\large{ \begin{cases}x + 2y < 30 \\ 3x+ y < 26 \end{cases}} \]

Let \( x \) and \( y \) be positive integers satisfying the system of inequality above.

Let the largest possible value of \( x \) be \( U \) and the largest possible value of \( y \) be \( V \).

(Obviously, if \( x = U \), \( y \) cannot simultaneously be equal to \( V \).)

Find \( U + V \)

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