Solve using RREF: Solve the system of equations using matrix. x1 + x3 = 5, x1 + x2 + 2x3 =-2, x1 +2x2 +3x3 =1

To solve the given system of equations using the Reduced Row Echelon Form (RREF), we represent the system as an augmented matrix and perform row operations. The system is:

\[
\begin{aligned}
1. \quad x_1 + x_3 &= 5 \\
2. \quad x_1 + x_2 + 2x_3 &= -2 \\
3. \quad x_1 + 2x_2 + 3x_3 &= 1
\]

### Step 1: Write the augmented matrix
\[
\begin{bmatrix}
1 & 0 & 1 & 5 \\
1 & 1 & 2 & -2 \\
1 & 2 & 3 & 1
\end{bmatrix}
\]

### Step 2: Perform row operations to reduce the matrix to RREF

I'll perform the computations to find the RREF. sympy as sp

# Define the augmented matrix
matrix = sp.Matrix([
    [1, 0, 1, 5],
    [1, 1, 2, -2],
    [1, 2, 3, 1]
])

# Perform row reduction to RREF
rref_matrix, pivots = matrix.rref()
rref_matrixThe Reduced Row Echelon Form (RREF) of the augmented matrix is:

\[
\begin{bmatrix}
1 & 0 & 1 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]

### Interpretation of the RREF
From the matrix, the system of equations becomes:
1. \( x_1 + x_3 = 0 \)
2. \( x_2 + x_3 = 0 \)
3. \( 0 = 1 \) (this is inconsistent)

Since the third row represents a contradiction (\(0 = 1\)), the system of equations is inconsistent, meaning there are no solutions.

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Would you like a detailed breakdown of each step? Here are some follow-up questions to consider:

1. What does it mean for a system of equations to be inconsistent?
2. How are row operations applied systematically in matrix reduction?
3. Can a system have infinitely many solutions? When does that occur?
4. How can we verify matrix calculations manually?
5. What are the differences between RREF and REF?

**Tip:** Always double-check for arithmetic errors when performing row operations manually!

Learn how to solve a system of linear equations using the Reduced Row Echelon Form (RREF) method. This example demonstrates the step-by-step process of transforming a matrix and interpreting the results. The system is inconsistent, meaning there are no solutions. Suitable for students in Grades 10-12 studying linear algebra.

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