Answer :
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.
[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.