Solution RREF.C17 Robert Beezer

We row-reduce the augmented matrix of the system of equations, \begin{align*} \begin{bmatrix} -1 & 5 & 0 & 0 & -8 \\ -2 & 5 & 5 & 2 & 9 \\ -3 & -1 & 3 & 1 & 3 \\ 7 & 6 & 5 & 1 & 30 \end{bmatrix} &\rref \begin{bmatrix} \leading{1} & 0 & 0 & 0 & 3 \\ 0 & \leading{1} & 0 & 0 & -1 \\ 0 & 0 & \leading{1} & 0 & 2 \\ 0 & 0 & 0 & \leading{1} & 5 \end{bmatrix} \end{align*} This augmented matrix represents the linear system $x_1=3$, $x_2=-1$, $x_3=2$, $x_4=5$, which clearly has only one possible solution. We can write this solution set then as \begin{align*} S&=\set{\colvector{3\\-1\\2\\5}} \end{align*}