For them to be dependent means that we can find \(\displaystyle \lambda _i\) such that
\(\displaystyle \lambda _1 v_1 + \lambda _2 v_2 + \lambda _3 v_3 = 0\)
where at least one of the \(\displaystyle \lambda _i \neq 0\).

The other, more efficient way, is to use matrices:
\(\displaystyle \left ( \begin{matrix} 1 & 1 & x \\ 2 & 1 & 6 \\ x & 1 & 2 \end{matrix} \right ) \cdot \left ( \begin{matrix} \lambda _1 \\ \lambda _2 \\ \lambda _3 \end{matrix} \right ) = \left ( \begin{matrix} 0\\ 0 \\ 0 \end{matrix} \right )\)

If the determinant of the coefficient matrix is not 0 then the system is linearly independent.

Notice that topsquark said "If the determinant of the coefficient matrix is not 0 then the system is linearly independent." You want linearly dependent.