Write the given matrix equation as a system of linear equations without matrices. begin{bmatrix}-1 & 0&1 0 & -1 &0 0&1&1 end{bmatrix}begin{bmatrix}x y z end{bmatrix}=begin{bmatrix}-4 2 4 end{bmatrix}

Write the given matrix equation as a system of linear equations without matrices.
$\left[\begin{array}{ccc}-1& 0& 1\\ 0& -1& 0\\ 0& 1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-4\\ 2\\ 4\end{array}\right]$
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Step 1
Given equation in matrix form:
$\left[\begin{array}{ccc}-1& 0& 1\\ 0& -1& 0\\ 0& 1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-4\\ 2\\ 4\end{array}\right]$
Step 2
Simplifying:
$\left[\begin{array}{ccc}-1& 0& 1\\ 0& -1& 0\\ 0& 1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-4\\ 2\\ 4\end{array}\right]$
$\left[\begin{array}{c}-1\cdot x+0\cdot y+1\cdot z\\ 0\cdot x-1\cdot y+0\cdot z\\ 0\cdot x+1\cdot y+1\cdot z\end{array}\right]=\left[\begin{array}{c}-4\\ 2\\ 4\end{array}\right]$
$\left[\begin{array}{c}-x+z\\ -y\\ y+z\end{array}\right]=\left[\begin{array}{c}-4\\ 2\\ 4\end{array}\right]$
Comparing both matrices:
The system of linear equations formed:
$-x+z=-4$
$-y=2$
$y+z=4$
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