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 $\displaystyle \left \{ (4,1,2,-1) \right \}$.