Answer
For all $h$ with $h\neq-15.$
Work Step by Step
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 $h\neq-15$.
