solving systems of linear equations:
For a system to have infinitely-many solutions, one of the lines in the system has to turn into all zeroes, after a sufficient number of row operations.
For a system to have no solution, one of the lines has to reduce to something non-sensical, like "2 = 0".
For a unique solution, you have to get single numerical values for each of x, y, and z.
(I don't know how your book is defining a "trivial" solution in this context, but it might mean "everything is zero".)
So do the row operations, trying to get the system into row-echelon or reduced-row-echelon form, and see where that leads!