# 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}

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

2021-02-27
Step 1
Given equation in matrix form:
$$\begin{bmatrix}-1 & 0&1 \\ 0 & -1 &0 \\ 0&1&1 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}=\begin{bmatrix}-4 \\ 2 \\ 4 \end{bmatrix}$$
Step 2
Simplifying:
$$\begin{bmatrix}-1 & 0&1 \\ 0 & -1 &0 \\ 0&1&1 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}=\begin{bmatrix}-4 \\ 2 \\ 4 \end{bmatrix}$$
$$\begin{bmatrix}-1 \cdot x+ 0\cdot y +1\cdot z \\ 0\cdot x -1 \cdot y + 0 \cdot z \\ 0 \cdot x + 1\cdot y +1 \cdot z \end{bmatrix}=\begin{bmatrix}-4 \\ 2 \\ 4 \end{bmatrix}$$
$$\begin{bmatrix}-x+z \\ -y \\ y+z \end{bmatrix}=\begin{bmatrix}-4 \\ 2 \\ 4 \end{bmatrix}$$
Comparing both matrices:
The system of linear equations formed:
$$-x+z=-4$$
$$-y=2$$
$$y+z=4$$

### Relevant Questions

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