Intermediate Algebra, Chapter 4, 4.2, Section 4.2, Problem 44

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

3x + y - z + 2w =& 9 \\
x + y + 2z - w =& 10 \\
x - y - z + 3w =& -2 \\
-x + y - z + w =& -6

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


$
\begin{equation}
\begin{aligned}

3x + y - z + 2w =& 9
&& \text{Equation 1}
\\
2x + 2y + 4z - 2w =& 20
&& 2 \times \text{ Equation 2}
\\
\hline

\end{aligned}
\end{equation}
$




$
\begin{equation}
\begin{aligned}

5x + 3y + 3z \phantom{-2w} =& 29
&& \text{Add; New equation 2}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

-9x - 3y + 3z - 6w =& -27
&& -3 \times \text{ Equation 1}
\\
2x - 2y - 2z+ 6w =& -4
&& 2 \times \text{ Equation 3}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

-7x - 5y + z \phantom{+6w} =& -31
&& \text{Add; New equation 3}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

3x + y - z + 2w =& 9
&& \text{Equation 1}
\\
2x -2y + 2z - 2w =& 12
&& -2 \times \text{ Equation 4}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

5x - y + z \phantom{-2w} =& 21
&& \text{Add; New Equation 4}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

5x + 3y + 3z =& 29
&& \text{Equation 2}
\\
21x + 15y - 3z =& 93
&& -3 \times \text{ Equation 3}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

26x + 18y \phantom{-3z} =& 122
&& \text{Add; New Equation 3}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

5x + 3y + 3z =& 29
&& \text{Equation 2}
\\
-15x + 3y - 3z =& -63
&& -3 \times \text{ Equation 4}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

-10x + 6y \phantom{-3z} =& -34
&& \text{Add; New Equation 4}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

26x + 18y =& 122
&& \text{Equation 3}
\\
30x - 18y =& 102
&& -3 \times \text{Equation 4}
\\
\hline

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

56x \phantom{-18y} =& 224
&& \text{Add}
\\
x =& 4
&& \text{Divide each side by $56$}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

-10(4) + 6y =& -34
&& \text{Substitute } x = 4 \text{ in New Equation 4}
\\
-40 + 6y =& -34
&& \text{Multiply}
\\
6y =& 6
&& \text{Add each side by $40$}
\\
y =& 1
&&

\end{aligned}
\end{equation}
$




$
\begin{equation}
\begin{aligned}

5(4) - 1 + z =& 21
&& \text{Substitute } x = 4 \text{ and } y = 1
\\
20 - 1 + z =& 21
&& \text{Multiply}
\\
19 + z =& 21
&& \text{Combine like terms}
\\
z =& 2
&& \text{Subtract each side by $19$}

\end{aligned}
\end{equation}
$



$
\begin{equation}
\begin{aligned}

3(4) + 1 - 2 + 2w =& 9
&& \text{Substitute } x = 4, y = 1 \text{ and } z = 2 \text{ in Equation 1}
\\
12 + 1 - 2 + 2w =& 9
&& \text{Multiply}
\\
11 + 2w =& 9
&& \text{Combine like terms}
\\
2w =& -2
&& \text{Subtract each side by $11$}
\\
w =& -1
&& \text{Divide each side by $2$}

\end{aligned}
\end{equation}
$


The solution set is $\displaystyle \left \{ (4,1,2,-1) \right \}$.