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x+y=11y+x=7\large \begin{aligned} x+\sqrt{y}&=11 \\ y+\sqrt{x}&=7 \end{aligned}x+yy+x=11=7
Let nnn be the number of real pairs (x,y)(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)(x1,y1),…,(xn,yn). Let
S=∑i=1n(xi+yi).S=\displaystyle \sum_{i=1}^n \left(x_i + y_i\right).S=i=1∑n(xi+yi).
Find n+Sn+Sn+S.
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