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Write the system of equations associated with the augmented matrix. Do not solve.

[tex]\[
\left[\begin{array}{lll|l}
1 & 0 & 0 & 2 \\
0 & 1 & 0 & 5 \\
0 & 0 & 1 & 1
\end{array}\right]
\][/tex]

Complete the system shown below. Use the letters \( x \), \( y \), and \( z \) to represent the terms from the first, second, and third columns of the matrix, respectively.

[tex]\[
\begin{cases}
x & = 2 \\
y & = 5 \\
z & = 1
\end{cases}
\][/tex]

(Simplify your answer.)



Answer :

GinnyAnswer
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.

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