Answer
$h=26$
Work Step by Step
The vectors are linearly dependent if and only if the matrix equation $\mathbf{A}\vec{x}=\vec{0}$ has a nontrivial solution, where $\mathbf{A}$ is the matrix whose columns are the given vectors. By row reduction, we conclude that $\begin{bmatrix}1&-5&1\\-1&7&1\\3&8&h\end{bmatrix}\sim\begin{bmatrix}1&-5&1\\0&1&1\\0&0&h-26\end{bmatrix}$.
Since the homogeneous matrix equation has a nontrivial solution if and only if at least one column of the coefficient matrix lacks a pivot, we set $h=26$, making the final row all zeros and providing a free variable in the third column.
