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find the augmented matrix and rref matrix and solution for each problem

Let's solve each system of equations by representing it in an augmented matrix form and then finding its reduced row echelon form (RREF). I'll begin by breaking down each problem.

### Problem 7
**Equations:**
1. \( 2x + 6y + z = 5 \)
2. \( x + y - z + 1 = 0 \) → rewrite as \( x + y - z = -1 \)
3. \( x + 8y + 2z = 0 \)

**Augmented Matrix Form:**
\[
\begin{bmatrix}
2 & 6 & 1 & | & 5 \\
1 & 1 & -1 & | & -1 \\
1 & 8 & 2 & | & 0
\end{bmatrix}
\]

### Finding the RREF:
Now, let's solve the augmented matrix using row operations to obtain the RREF.

1. **Row 1: \( [2, 6, 1 | 5] \) → Divide by 2:**
   \[
   [1, 3, 0.5 | 2.5]
   \]

2. **Make the leading 1 in the second row:**
   - Row 2 is already leading with \(x + y - z = -1\).
   - We need to make \(R2 = [0, -7, -1.5 | -3.5]\)



Find the augmented matrix and RREF matrix, and solution for each problem in the 3x3 systems of equations worksheet.

This page explains how to solve 3x3 systems of equations using augmented matrices and row reduction to RREF. Detailed solutions are provided for each system, covering key linear algebra concepts, including matrix operations and Gauss-Jordan elimination. Suitable for students in Grades 10-12.

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