Simultaneous Radical Equations!

Algebra Level 5

$\large \begin{aligned} x+\sqrt{y}&=11 \\ y+\sqrt{x}&=7 \end{aligned}$

Let $n$ be the number of real pairs $(x, y)$ to the above system of equations. Define these solution pairs as $(x_1, y_1), \ldots, (x_n, y_n)$ . Let

$S=\displaystyle \sum_{i=1}^n \left(x_i + y_i\right).$

Find $n+S$ .