## Example: Solving a 3-by-3 Matrix Equation

Solve `Ax = b` where:
```
A = [1  2  1]    b = [ 4]
    [2  5 -1]        [ 7]
    [1  1  2]        [ 5]
```

**Step 1:** Form the augmented matrix `[A | b]`
```
[ 1  2  1 |  4]
[ 2  5 -1 |  7]
[ 1  1  2 |  5]
```

**Step 2:** R2 = R2 - 2*R1, R3 = R3 - R1
```
[ 1  2  1 |  4]
[ 0  1 -3 | -1]
[ 0 -1  1 |  1]
```

**Step 3:** R3 = R3 + R2
```
[ 1  2  1 |  4]
[ 0  1 -3 | -1]
[ 0  0 -2 |  0]
```

**Step 4:** From R3: -2z = 0, so z = 0.
From R2: y - 3(0) = -1, so y = -1.
From R1: x + 2(-1) + 0 = 4, so x = 6.

**Solution:** x = [6, -1, 0]^

**Verification:**
- Row 1: 1(6) + 2(-1) + 1(0) = 6 - 2 = 4 ✓
- Row 2: 2(6) + 5(-1) + (-1)(0) = 12 - 5 = 7 ✓
- Row 3: 1(6) + 1(-1) + 2(0) = 6 - 1 = 5 ✓