## Linear Algebra and Its Applications (5th Edition)

For all $h$ with $h\neq-15.$
We know the system of equations is consistent if the last column of the corresponding augmented matrix is not a pivot column. So we first need to convert the matrix, $\left[ \begin{array}{ccc} 1 & -3 & -2\\ 5 & h & -7 \end{array} \right]$, to echelon form. We replace row 2 with -5(row 1)+(row 2) to get the equivalent matrix $$\left[ \begin{array}{ccc} 1 & -3 & -2\\ 0 & 15+h & 3 \end{array} \right].$$ Now, in order for the last column not to be a pivot column, we must have$$15+h\neq 0.$$ Hence, we must have $$h\neq -15.$$ Thus the corresponding system of equations is consistent for all $h\neq-15$.