#### Answer

$h=\frac{7}{2}$

#### 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, $\left[
\begin{array}{ccc}
2 & 3 & h\\
4 & 6 & 7
\end{array}
\right]$, to echelon form. We replace row 2 with -2(row 1)+(row 2) to get the equivalent matrix
$$\left[
\begin{array}{ccc}
2 & 3 & h\\
0 & 0 & -2h+7
\end{array}
\right].$$
Now, in order for the last column not to be a pivot column, we must have$$-2h+7=0.$$
We solve this equation to get $h=\frac{7}{2}.$
Hence if $h=\frac{7}{2}$, then the last column of the augmented matrix is not a pivot column, which means the corresponding system of equations is consistent.