Title: Using Gaussian Elimination to Determine If a System Has a Solution

This post explains how to use Gaussian Elimination to test whether a system of linear equations is consistent, inconsistent, or has infinite solutions. It walks through the full process from the augmented matrix to the final conclusion, showing how even one row can prove there's no solution.

Given System:

x - 2y + 3z = -2  
2x - y + z = 2  
2x - 7y + 11z = -9

Augmented matrix:

[ 1  -2   3  | -2 ]  
[ 2  -1   1  |  2 ]  
[ 2  -7  11  | -9 ]

Step 1: Use Row Operations to Reach Row-Echelon Form

Apply Gaussian Elimination to zero out lower values:

• Multiply row 1 and subtract from rows 2 and 3

• Resulting matrix:

[ 1  -2   3  | -2 ]  
[ 0   3  -5  |  6 ]  
[ 0  -3   5  | -5 ]

Add row 2 and 3 to eliminate the leading term in row 3:

[ 1  -2   3  | -2 ]  
[ 0   3  -5  |  6 ]  
[ 0   0   0  |  1 ]

Step 2: Interpret the Final Matrix

The third row says:

0x + 0y + 0z = 1

That is a contradiction → no solution

This system is inconsistent.

Visual Summary of Matrix Solution Types:

• If last row is [ 0 0 0 | 1 ] → no solution (inconsistent)

• If last row is [ 0 0 0 | 0 ] → infinite solutions (if others are consistent)

• If matrix reduces cleanly → unique solution

Final Answer:

No solution.
The system is inconsistent based on the Gaussian elimination result.

This walkthrough is from Chapter 8 of the Ultimate Crash Course for STEM Majors and is designed to help you recognize how and why systems fail before wasting time solving.

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