College Algebra, Chapter 7, Review Exercises, Section Review Exercises, Problem 20

Find the complete solution of the system
$
\left\{
\begin{equation}
\begin{aligned}

x - y - 2z + 3w =& 0
\\
y - z + w =& 1
\\
3x - 2y - 7z + 10w =& 2

\end{aligned}
\end{equation}
\right.
$
using Gauss-Jordan Elimination.

We transform the system into reduced row-echelon form

$\displaystyle \left[
\begin{array}{ccccc}
1 & -1 & -2 & 3 & 0 \\
0 & 1 & -1 & 1 & 1 \\
3 & -2 & -7 & 10 & 2
\end{array}
\right]$

$R_3 - 3 R_1 \to R_3$

$\displaystyle \left[
\begin{array}{ccccc}
1 & -1 & -2 & 3 & 0 \\
0 & 1 & -1 & 1 & 1 \\
0 & 1 & -1 & 1 & 2
\end{array}
\right]$

$R_3 - R_2 \to R_3$

$\displaystyle \left[
\begin{array}{ccccc}
1 & -1 & -2 & 3 & 0 \\
0 & 1 & -1 & 1 & 1 \\
0 & 0 & 0 & 0 & 1
\end{array}
\right]$

$R_2 - R_3 \to R_2$

$\displaystyle \left[
\begin{array}{ccccc}
1 & -1 & -2 & 3 & 0 \\
0 & 1 & -1 & 1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{array}
\right]$

$R_1 + R_2 \to R_1$

$\displaystyle \left[
\begin{array}{ccccc}
1 & 0 & -3 & 4 & 0 \\
0 & 1 & -1 & 1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{array}
\right]$


This is in reduced row echelon form. If we translate the last row back into equation, we get $0x + 0y + 0z + 0w = 1$, or $0 = 1$, which is false. This that the system has no solution or it is inconsistent.