College Algebra, Chapter 7, 7.1, Section 7.1, Problem 32

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

x - 2y + 5z =& 3
\\
-2x + 6y - 11z =& 1
\\
3x - 16y - 20z =& -26

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

We transform the system into row-echelon form.

$\left[ \begin{array}{cccc}
1 & -2 & 5 & 3 \\
-2 & 6 & -11 & 1 \\
3 & -16 & -20 & -26
\end{array} \right]$

$R_3 - 3R_1 \to R_3$

$\left[ \begin{array}{cccc}
1 & -2 & 5 & 3 \\
-2 & 6 & -11 & 1 \\
0 & -10 & -35 & -35
\end{array} \right]$

$R_2 + 2R_1 \to R_2$

$\left[ \begin{array}{cccc}
1 & -2 & 5 & 3 \\
0 & 2 & -1 & 7 \\
0 & -10 & -35 & -35
\end{array} \right]$

$\displaystyle R_3 + 5R_2 \to R_3$

$\left[ \begin{array}{cccc}
1 & -2 & 5 & 3 \\
0 & 2 & -1 & 7 \\
0 & 0 & -40 & 0
\end{array} \right]$

$\displaystyle \frac{-1}{40} R_3$

$\left[ \begin{array}{cccc}
1 & -2 & 5 & 3 \\
0 & 2 & -1 & 7 \\
0 & 0 & 1 & 0
\end{array} \right]$

$\displaystyle \frac{1}{2} R_2$

$\left[ \begin{array}{cccc}
1 & -2 & 5 & 3 \\
0 & 1 & \displaystyle \frac{-1}{2} & \displaystyle \frac{7}{2} \\
0 & 0 & 1 & 0
\end{array} \right]$

$R_1 + 2R_2 \to R_1$

$\left[ \begin{array}{cccc}
1 & 0 & 4 & 10 \\
0 & 1 & \displaystyle \frac{-1}{2} & \displaystyle \frac{7}{2} \\
0 & 0 & 1 & 0
\end{array} \right]$

$\displaystyle R_2 + \frac{1}{2} R_3 \to R_2$

$\left[ \begin{array}{cccc}
1 & 0 & 4 & 10 \\
0 & 1 & 0 & \displaystyle \frac{7}{2} \\
0 & 0 & 1 & 0
\end{array} \right]$

$R_1 - 4R_3 \to R_1$

$\left[ \begin{array}{cccc}
1 & 0 & 0 & 10 \\
0 & 1 & 0 & \displaystyle \frac{7}{2} \\
0 & 0 & 1 & 0
\end{array} \right]$

We now have an equivalent matrix in reduced row-echelon form and the corresponding system of equations is


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

x =& 10
\\
y =& \frac{7}{2}
\\
z =& 0

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