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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