Answer
No, this system has no nontrivial solutions.
Work Step by Step
To solve this problem, we perform elimination on the coefficient matrix of the system:
$\begin{bmatrix}1&-3&7\\-2&1&-4\\1&2&9\end{bmatrix}$
(1) Add $2$ times row 1 to row 2, and add $-1$ times row 1 to row 3.
$\begin{bmatrix}1&-3&7\\0&-5&10\\0&5&2\end{bmatrix}$
(2) Add row 2 to row 3.
$\begin{bmatrix}1&-3&7\\0&-5&10\\0&0&12\end{bmatrix}$
Without further computation, we can see that every column of the coefficient matrix contains a pivot. Hence, the solution set involves no free variables, so the homogeneous equation has only the trivial solution. (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.)
