If the vectors are linearly dependent there will be a set of coefficients, not all zero, such that

This you can translate into a set of three linear equations.

For non-trivial solutions, it is necessary that the determinant of coefficients is zero, and this you have demonstrated, in which case the vectors are indeed linearly dependent.

Easier though is to simply solve the equations using elementary row operations. I get

where is some arbitrary number.

(If the equations turn out to be inconsistent, it means that the vectors are linearly independent.)