Intermediate Algebra, Chapter 4, 4.2, Section 4.2, Problem 46

Solve each system $
\begin{equation}
\begin{aligned}

x - 3y + 7z + w =& 11 \\
2x + 4y + 6z - 3w =& -3 \\
3x + 2y + z + 2w =& 19 \\
4x + y - 3z + w =& 22

\end{aligned}
\end{equation}
$ by expressing the solution in the form $(x,y,z,w)$.


$
\begin{equation}
\begin{aligned}

3x - 9y + 21z + 3w =& 33
&& 3 \times \text{ Equation 1}
\\
2x + 4y + 6z - 3w =& -3
&& \text{Equation 2}
\\
\hline

\end{aligned}
\end{equation}
$




$
\begin{equation}
\begin{aligned}

5x - 5y + 27z \phantom{-3w} =& 30
&& \text{Add; New equation 2}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

-2x + 6y - 14z - 2w =& -22
&& -2 \times \text{ Equation 1}
\\
3x + 2y + z + 2w =& 19
&& \text{Equation 3}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

x + 8y - 13z \phantom{+2w} =& -3
&& \text{Add; New equation 3}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

-x + 3y - 7z - w =& -11
&& -1 \times \text{ Equation 1}
\\
4x + y - 3z + w =& 22
&& \text{Equation 4}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

3x + 4y - 10z \phantom{+w} =& 11
&& \text{Add; New Equation 4}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

65x - 65y + 351z =& 390
&& 13 \times \text{ New Equation 2}
\\
27x + 216y - 315z =& -81
&& 27 \times \text{ New Equation 3}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

92x + 151y \phantom{-351z} =& 309
&& \text{Add; New Equation 3}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

50x - 50y + 270z =& 300
&& 10 \times \text{ New Equation 2}
\\
81x + 108y - 270z =& 297
&& 27 \times \text{ New Equation 4}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

131x + 58y \phantom{-270z} =& 597
&& \text{Add; New Equation 4}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

92x + 151y =& 309
&& \text{Equation 3}
\\
131x + 58y =& 597
&& \text{Equation 4}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

-5336x - 8758y =& -17922
&& -58 \times \text{ Equation 3}
\\
19781x + 8758y =& 90147
&& 151 \times \text{ Equation 4}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

14445x \phantom{+8758y} =& 72225
&& \text{Add}
\\
x =& 5
&& \text{Divide each side by $14445$}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

92(5) + 151y =& 309
&& \text{Substitute } x = 5 \text{ in New Equation 3}
\\
460 + 151y =& 309
&& \text{Multiply}
\\
151y =& -151
&& \text{Subtract each side by $460$}
\\
y =& -1
&& \text{Divide each side by $151$}

\end{aligned}
\end{equation}
$




$
\begin{equation}
\begin{aligned}

5 + 8(-1) - 13z =& -3
&& \text{Substitute } x = 5 \text{ and } y = -1 \text{ in Equation 1}
\\
5 - 8 - 13z =& -3
&& \text{Multiply}
\\
-3 - 13z =& 0
&& \text{Combine like terms}
\\
-13z =& 2
&& \text{Add each side by $3$}
\\
z =& 0
&& \text{Divide each side by $-13$}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

5 - 3(-1) + 7(0) + w =& 11
&& \text{Substitute } x = 5, y = -1 \text{ and } z = 0 \text{ in Equation 1}
\\
5 + 3 + 0 + w =& 11
&& \text{Multiply}
\\
8 + w =& 11
&& \text{Combine like terms}
\\
w =& 3
&& \text{Subtract each side by $8$}

\end{aligned}
\end{equation}
$


The solution set is $\displaystyle \left \{ (5,-1,0,3) \right \}$.