College Algebra, Chapter 7, Review Exercises, Section Review Exercises, Problem 12

Find the complete solution of the system
$
\left\{
\begin{array}{cccccc}
x & & +3z & & = & -1 \\
& y & & -4w & = & 5 \\
& 2y & +z & +w & = & 0 \\
2x & +y & +5z & -4w & = & 4
\end{array}
\right.
$
using Gaussian Elimination.

For this system we have

$\displaystyle \left[
\begin{array}{ccccc}
1 & 0 & 3 & 0 & -1 \\
0 & 1 & 0 & -4 & 5 \\
0 & 2 & 1 & 1 & 0 \\
2 & 1 & 5 & -4 & 4
\end{array}
\right]$

$R_4 - 2R_1 \to R_4$

$\displaystyle \left[
\begin{array}{ccccc}
1 & 0 & 3 & 0 & -1 \\
0 & 1 & 0 & -4 & 5 \\
0 & 2 & 1 & 1 & 0 \\
0 & 1 & -1 & -4 & 6
\end{array}
\right]$

$R_3 - 2 R_2 \to R_3$

$\displaystyle \left[
\begin{array}{ccccc}
1 & 0 & 3 & 0 & -1 \\
0 & 1 & 0 & -4 & 5 \\
0 & 0 & 1 & 9 & -10 \\
0 & 1 & -1 & -4 & 6
\end{array}
\right]$

$R_4 - R_2 \to R_4$

$\displaystyle \left[
\begin{array}{ccccc}
1 & 0 & 3 & 0 & -1 \\
0 & 1 & 0 & -4 & 5 \\
0 & 0 & 1 & 9 & -10 \\
0 & 0 & -1 & 0 & 1
\end{array}
\right]$

$R_4 + R_3 \to R_4$

$\displaystyle \left[
\begin{array}{ccccc}
1 & 0 & 3 & 0 & -1 \\
0 & 1 & 0 & -4 & 5 \\
0 & 0 & 1 & 9 & -10 \\
0 & 0 & 0 & 9 & -9
\end{array}
\right]$

$\displaystyle \frac{1}{9} R_4$

$\displaystyle \left[
\begin{array}{ccccc}
1 & 0 & 3 & 0 & -1 \\
0 & 1 & 0 & -4 & 5 \\
0 & 0 & 1 & 9 & -10 \\
0 & 0 & 0 & 1 & -1
\end{array}
\right]
$


The corresponding system is


$
\left\{
\begin{array}{cccccc}
x & & +3z & & = & -1 \\
& y & & -4w & = & 5 \\
& & z & +9w & = & -10 \\
& & & w & = & -1
\end{array}
\right.
$


We now use back-substitution


$
\begin{equation}
\begin{aligned}

z + 9(-1) =& -10
&& \text{Back-substitute } w = -1
\\
z =& -10 + 9
&& \text{Subtract } 9(-1) = -9
\\
z =& -1
&&
\\
\\
y - 4 (-1) =& 5
&& \text{Back-substitute } w = -1
\\
y =& 5-4
&& \text{Subtract } -4(-1)=4
\\
y =& 1
&&
\\
\\
x + 3(-1) =& -1
&& \text{Back-substitute } z = -1
\\
x =& -1+3
&& \text{Subtract } 3(-1) = -3
\\
x =& 2
&&


\end{aligned}
\end{equation}
$


The solution is


$
\begin{equation}
\begin{aligned}

x =& 2
\\
y =& 1
\\
z =& -1
\\
w =& -1

\end{aligned}
\end{equation}
$