7. Consider the following system of equations over \( \mathbb{Z}_{5} \) : \[ \begin{array}{l} x+2 y+w=1 \\ 2 x+y+z=2 \\ x+y+2 z+2 w=1 \\ \end{array} \] Working over \( \mathbb{Z}_{5} \), how many distinct solutions are there for \( (x, y, z, w) \) ? no solutions exactly twenty-five infinitely many exactly one exactly five
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The system of equations is: \[ \begin{array}{l} x + 2y + w = 1 \\ 2x + y + z = 2 \\ x + y + 2z + 2w = 1 \\ \end{array} \] In matrix form, this can be written as: \[ \begin{pmatrix} 1 & 2 & 0 & 1 \\ 2 & 1 & 1 & 0 \\ 1 & 1 & 2 & 2 Show more…
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