https://file.aiquickdraw.com/mathbot/file/1729009930468t1bi7w86.jpg

Let's analyze the matrix provided in the image. The matrix is:

\[
\begin{pmatrix}
1 & 0 & -2 & -8.5 \\
4 & 2 & 0 & 10 \\
2 & 0 & -3 & -12
\end{pmatrix}
\]

This is a system of linear equations in the form \(Ax = b\), and we need to solve for \(x\), \(y\), and \(z\).

### Step 1: Convert the matrix to Row Echelon Form (REF)

We already have a leading 1 in the first row, so we start by eliminating the entries below it.

#### Row operations:

- **Row 2**: Subtract 4 times **Row 1** from **Row 2**:
\[
R2 = R2 - 4R1
\]
\[
R2 = [4 \ 2 \ 0 \ 10] - 4 \times [1 \ 0 \ -2 \ -8.5]
= [0 \ 2 \ 8 \ 44]
\]

- **Row 3**: Subtract 2 times **Row 1** from **Row 3**:
\[
R3 = R3 - 2R1
\]
\[
R3 = [2 \ 0 \ -3 \ -12] - 2 \times [1 \ 0 \ -2 \ -8.5] 
= [0 \ 0 \ 1 \ 5]
\]

Thus, after these operations, the matrix becomes:
\[
\begin{pmatrix}
1 & 0 & -2 & -8.5 \\
0 & 2 & 8 & 44 \\
0 & 0 & 1 & 5
\end{pmatrix}
\]

### Step 2: Make further eliminations

Now we focus on simplifying **Row 2** by eliminating the coefficient in front of \(z\) in **Row 1**.

- **Row 1**: Add 2 times **Row 3** to **Row 1** to eliminate the \(-2\) in the first row, third column:
\[
R1 = R1 + 2R3
\]
\[
R1 = [1 \ 0 \ -2 \ -8.5] + 2 \times [0 \ 0 \ 1 \ 5] = [1 \ 0 \ 0 \ 1.5]
\]

At this stage, the matrix looks like this:
\[
\begin{pmatrix}
1 & 0 & 0 & 1.5 \\
0 & 2 & 8 & 44 \\
0 & 0 & 1 & 5
\end{pmatrix}
\]

Finally, we simplify **Row 2** by eliminating the 8 in front of \(z\):
\[
R2 = R2 - 8R3
\]
\[
R2 = [0 \ 2 \ 8 \ 44] - 8 \times [0 \ 0 \ 1 \ 5] = [0 \ 2 \ 0 \ 4]
\]

Divide **Row 2** by 2:
\[
R2 = [0 \ 1 \ 0 \ 2]
\]

### Final Matrix (Reduced Row Echelon Form):

\[
\begin{pmatrix}
1 & 0 & 0 & 1.5 \\
0 & 1 & 0 & 2 \\
0 & 0 & 1 & 5
\end{pmatrix}
\]

### Solution:
From this matrix, we can directly read off the values of \(x\), \(y\), and \(z\):
\[
x = 1.5, \quad y = 2, \quad z = 5
\]

Would you like more details about the steps involved? Here are some related questions:

1. How do we solve a system of linear equations using Gaussian elimination?
2. What are the conditions for a matrix to

To solve a system of linear equations relating x, y, and z, Silas writes the following matrix. After converting the matrix to row echelon form, Silas uses the results to solve the original system of equations. Select the correct choices to complete the statement about Silas’s solutions.

Learn how to solve a system of linear equations using Gaussian elimination and row echelon form. The problem involves converting a matrix and applying row operations to find the solution for x, y, and z. Detailed steps and solution provided. Suitable for high school students in Grades 10-12.

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