If $v_{1}=(1,1,1) ; v_{2}=(0,1,1) ; v_{3}=(0, x, 1)$ are linearly dependent vectors in $R^{3}$ find the value for $x_{6} ?$

Step 1 ：Form the matrix with the given vectors as columns and find its determinant: $\begin{vmatrix} 1 & 0 & 0 \\ 1 & 1 & x \\ 1 & 1 & 1 \end{vmatrix}$

Step 2 ：Calculate the determinant: $(1)(1-1) - (0)(1-x) + (0)(x-1) = 0$

Step 3 ：\(\boxed{x = 1}\)