College Algebra, Chapter 7, 7.1, Section 7.1, Problem 34

State whether the system of linear equations $\left\{ \begin{equation}
\begin{aligned}

-2x + 6y - 2z =& -12
\\
x - 3y + 2z =& 10
\\
-x + 3y + 2z =& 6

\end{aligned}
\end{equation} \right.$ is inconsistent or dependent. If it is dependent, find the complete solution.

We can write the system into simplest form


$
\left\{
\begin{equation}
\begin{aligned}

-x + 3y - z =& -6
\\
x - 3y + 2z =& 10
\\
-x + 3y + 2z =& 6

\end{aligned}
\end{equation}
\right.
$


We transform the system into row-echelon form.

$\left[ \begin{array}{cccc}
-1 & 3 & -1 & -6 \\
1 & -3 & 2 & 10 \\
-1 & 3 & 2 & 6
\end{array} \right]$

$-R_1$

$\left[ \begin{array}{cccc}
1 & -3 & 1 & 6 \\
1 & -3 & 2 & 10 \\
-1 & 3 & 2 & 6
\end{array} \right]$

$R_3 + R_1 \to R_3$

$\left[ \begin{array}{cccc}
1 & -3 & 1 & 6 \\
1 & -3 & 2 & 10 \\
0 & 0 & 3 & 12
\end{array} \right]$

$\displaystyle \frac{1}{3} R_3$

$\left[ \begin{array}{cccc}
1 & -3 & 1 & 6 \\
1 & -3 & 2 & 10 \\
0 & 0 & 1 & 4
\end{array} \right]$


$R_2 - R_1 \to R_2$

$\left[ \begin{array}{cccc}
1 & -3 & 1 & 6 \\
0 & 0 & 1 & 4 \\
0 & 0 & 1 & 4
\end{array} \right]$


The matrix has infinitely many solutions to obtain the complete solution, we let $t$ represent any real number, we expresses $x$ and $y$ in terms of $t$.


$
\begin{equation}
\begin{aligned}

x =& 3t + 6 - z
\\
=& 3t + 6 - 4
\\
=& 3t + 2
\\
y =& t
\\
z =& 4

\end{aligned}
\end{equation}
$


We can also write the solution as the ordered triple $(3t + 2, t, 4)$, where $t$ is any real number.