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