Certainly! Let's look at the augmented matrix provided:
[tex]\[
\left[\begin{array}{lll|l}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 5 \\
0 & 0 & 1 & 1
\end{array}\right]
\][/tex]
Each row of the matrix represents an equation in a system of linear equations.
### First Row:
[tex]\[
1 \cdot x + 0 \cdot y + 0 \cdot z = 2
\][/tex]
Simplifying this equation, we get:
[tex]\[
x = 2
\][/tex]
### Second Row:
[tex]\[
0 \cdot x + 1 \cdot y + 0 \cdot z = 5
\][/tex]
Simplifying this equation, we get:
[tex]\[
y = 5
\][/tex]
### Third Row:
[tex]\[
0 \cdot x + 0 \cdot y + 1 \cdot z = 1
\][/tex]
Simplifying this equation, we get:
[tex]\[
z = 1
\][/tex]
Putting it all together, the system of equations associated with the augmented matrix is:
[tex]\[
\begin{cases}
x = 2 \\
y = 5 \\
z = 1
\end{cases}
\][/tex]
So, filling in the blanks:
$
x = 2
$
$
y = 5
$
$
z = 1
$
These are the equations derived from the given augmented matrix.
