3 Equations, 4 Variables, \(\infty \) Solutions

Algebra Level 3

\[ \left[ \begin{array} {cccc} 2 & 4 & 0 & 1 \\ 5 & 2 & 1 & 2 \\ 3 & 3 & 1 & 1 \end{array} \right] \left[\begin{array} {c} w \\ x \\ y \\ z \\ \end{array} \right] =\left[ \begin{array} {c} 7 \\ 0 \\ 8 \\ \end{array} \right] \]

The system of linear equations shown above has infinite solutions. One of the four variables is fixed, that is over all the solutions to the above system, the fixed variable takes only one value.

Which is the fixed variable?

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