Simultaneous Radical Equations!

Algebra Level 5

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

Let nn be the number of real pairs (x,y)(x, y) to the above system of equations. Define these solution pairs as (x1,y1),,(xn,yn)(x_1, y_1), \ldots, (x_n, y_n). Let

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

Find n+Sn+S.

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