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nxn system x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n

Source: 1979 Swedish Mathematical Competition p1

March 26, 2021
algebrasystem of equationsSystem

Problem Statement

Solve the equations: {x1+2x2+3x3+⋯+(n−1)xn−1+nxn=n2x1+3x2+4x3+⋯+nxn−1+xn=n−13x1+4x2+5x3+⋯+xn−1+2xn=n−2⋯⋯⋯⋯⋅⋯⋯⋯⋯⋅⋯⋯⋯⋯⋅(n−1)x1+nx2+x3+⋯+(n−3)xn−1+(n−2)xn=2nx1+x2+2x3+⋯+(n−2)xn−1+(n−1)xn=1\left\{ \begin{array}{l} x_1 + 2 x_2 + 3 x_3 + \cdots + (n-1) x_{n-1} + n x_n = n \\ 2 x_1 + 3 x_2 + 4 x_3 + \cdots + n x_{n-1} + x_n = n-1 \\ 3 x_1 + 4 x_2 + 5 x_3 + \cdots + x_{n-1} + 2 x_n = n-2 \\ \cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot\cdots \cdots \cdots \cdots\cdot \\ (n-1) x_1 + n x_2 + x_3 + \cdots + (n-3) x_{n-1} + (n-2) x_n = 2 \\ n x_1 + x_2 + 2 x_3 + \cdots + (n-2) x_{n-1} + (n-1) x_n = 1 \end{array} \right.
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