https://file.aiquickdraw.com/mathbot/file/1726779619052k5l5ha5r.jpg

I see two systems of linear equations on the uploaded image. Let's solve both using matrix methods.

### First System of Equations:
The first system of equations is:
\[
2x - y - z = -10 \tag{1}
\]
\[
-x + 2y + z = 3 \tag{2}
\]
\[
x + 2y + 3z = 13 \tag{3}
\]

This can be represented as an augmented matrix:
\[
\begin{bmatrix}
2 & -1 & -1 & | & -10 \\
-1 & 2 & 1 & | & 3 \\
1 & 2 & 3 & | & 13
\end{bmatrix}
\]

We will reduce this matrix to Row Echelon Form (REF) or Reduced Row Echelon Form (RREF) and solve for \(x\), \(y\), and \(z\).

### Second System of Equations:
The second system of equations is:
\[
0.2x + 5y - 3z = 14 \tag{4}
\]
\[
-x - 2y + 4z = -12 \tag{5}
\]
\[
-x + 3y - 2z = 13 \tag{6}
\]

This can also be represented as an augmented matrix:
\[
\begin{bmatrix}
0.2 & 5 & -3 & | & 14 \\
-1 & -2 & 4 & | & -12 \\
-1 & 3 & -2 & | & 13
\end{bmatrix}
\]

Again, we will reduce this matrix to solve for \(x\), \(y\), and \(z\).

Let me solve both systems of equations using matrix row reduction.The solutions for both systems of linear equations have been found by reducing the augmented matrices to their Reduced Row Echelon Form (RREF).

### First System of Equations:
The RREF for the first system is:
\[
\begin{bmatrix}
1 & 0 & 0 & | & -3 \\
0 & 1 & 0 & | & -4 \\
0 & 0 & 1 & | & 8
\end{bmatrix}
\]
This corresponds to:
\[
x = -3, \quad y = -4, \quad z = 8
\]

### Second System of Equations:
The RREF for the second system is:
\[
\begin{bmatrix}
1 & 0 & 0 & | & -3.494 \\
0 & 1 & 0 & | & 0.880 \\
0 & 0 & 1 & | & -3.434
\end{bmatrix}
\]
This corresponds to:
\[
x \approx -3.494, \quad y \approx 0.880, \quad z \approx -3.434
\]

Would you like more details or further clarification?

Here are 5 related questions:
1. How do you interpret a system of equations with no solutions?
2. What are some practical applications of solving systems of linear equations?
3. Can you explain the difference between Row Echelon Form and Reduced Row Echelon Form?
4. How does Gaussian elimination differ from the RREF method?
5. What happens when a system has infinitely many solutions?

**Tip**: Always check if a system is consistent or inconsistent before solving, as this affects the number of solutions.

Solve the two systems of linear equations represented by augmented matrices.

This problem involves solving two systems of linear equations using augmented matrices. Learn how to reduce matrices to Row Echelon Form (REF) and Reduced Row Echelon Form (RREF) using Gaussian Elimination.

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