Solution RREF.C13 Robert Beezer

The augmented matrix of the system of equations is \begin{equation*} \begin{bmatrix} 1 & 2 & 8 & -7 & -2\\ 3 & 2 & 12 & -5 & 6\\ -1 & 1 & 1 & -5 & -10 \end{bmatrix} \end{equation*} which row-reduces to \begin{equation*} \begin{bmatrix} \leading{1} & 0 & 2 & 1 & 0\\ 0 & \leading{1} & 3 & -4 & 0\\ 0 & 0 & 0 & 0 & \leading{1} \end{bmatrix} \end{equation*} Row 3 represents the equation $0=1$, which is patently false, so the original system has no solutions. We can express the solution set as the empty set, $\emptyset=\set{\ }$.