To write the system of linear equations represented by the given augmented matrix, we will translate each row into an equation. The variables used will be [tex]\( w, x, y, \)[/tex] and [tex]\( z \)[/tex].
Given the augmented matrix:
[tex]\[
\left[\begin{array}{rrrr|r}
1 & -2 & 4 & 3 & 10 \\
0 & 1 & 4 & 0 & 14 \\
0 & 0 & 1 & -1 & 3 \\
1 & -2 & 0 & 0 & -2
\end{array}\right]
\][/tex]
### First Row:
The first row is [tex]\([1, -2, 4, 3 | 10]\)[/tex]. This translates to the linear equation:
[tex]\[
1w - 2x + 4y + 3z = 10
\][/tex]
So the equation represented by the first row is:
[tex]\[
w - 2x + 4y + 3z = 10
\][/tex]
### Second Row:
The second row is [tex]\([0, 1, 4, 0 | 14]\)[/tex]. This translates to the linear equation:
[tex]\[
0w + 1x + 4y + 0z = 14
\][/tex]
Simplifying this, we get:
[tex]\[
x + 4y = 14
\][/tex]
### Third Row:
The third row is [tex]\([0, 0, 1, -1 | 3]\)[/tex]. This translates to the linear equation:
[tex]\[
0w + 0x + 1y - 1z = 3
\][/tex]
Simplifying this, we get:
[tex]\[
y - z = 3
\][/tex]
### Fourth Row:
The fourth row is [tex]\([1, -2, 0, 0 | -2]\)[/tex]. This translates to the linear equation:
[tex]\[
1w - 2x + 0y + 0z = -2
\][/tex]
Simplifying this, we get:
[tex]\[
w - 2x = -2
\][/tex]
Therefore, the system of linear equations represented by the given augmented matrix is:
[tex]\[
\begin{cases}
w - 2x + 4y + 3z = 10 \\
x + 4y = 14 \\
y - z = 3 \\
w - 2x = -2
\end{cases}
\][/tex]
