Answer
Yes, the system has nontrivial solutions.
Work Step by Step
We solve by elimination on the coefficient matrix of the linear system:
$\begin{bmatrix}2&-5&8\\-2&-7&1\\4&2&7\end{bmatrix}$
(1) Add row 1 to row 2, and add $-2$ times row 1 to row 3.
$\begin{bmatrix}2&-5&8\\0&-12&9\\0&12&-9\end{bmatrix}$
(2) Add row 2 to row 3.
$\begin{bmatrix}2&-5&8\\0&-12&9\\0&0&0\end{bmatrix}$
Without further computation, we can see that the third column of the coefficient matrix lacks a pivot, meaning the solution set involves a free variable. Therefore, the homogeneous linear system has nontrivial solutions. (Note that, since the final column of the augmented matrix is all zeros, we need examine only the coefficient matrix. This is not the case for non-homogeneous systems.)
