Squares Galore

Algebra Level 5

\[\begin{array} { r r r r r r r r l} x_1 &+ 4x_2 &+ 9x_3 &+ 16x_4 &+25x_5 &+ 36x_6& + 49x_7 &=& 1 \\ 4x_1 &+ 9x_2 &+ 16x_3 &+ 25x_4 &+ 36x_5 &+ 49x_6 &+ 64x_7 &=& 10 \\ 16x_1 &+ 25x_2 &+ 36x_3 &+ 49x_4 &+ 64x_5 &+ 81x_6 &+ 100x_7 &=& 100 \\ \end{array}\]

Given that \(x_1\), \(x_2\), \(\ldots\) \(x_7\) are real numbers that satisfy the system of equations above.

What is the value of \(9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7\)?

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