If $v_{1}=(1,0,3) ; v_{2}=(0,2,4) ; v_{3}=(3, x, 1)$ are linearly dependent vectors in $R^{3}$ find the value for $x$ ? (A) 6 (B) 1 (C) 5 (D) 0 (E) -4

Step 1 ：Since the vectors are linearly dependent, the determinant of the matrix formed by these vectors should be equal to 0. Calculate the determinant: \(\begin{vmatrix} 1 & 0 & 3 \\ 0 & 2 & x \\ 3 & 4 & 1 \end{vmatrix}\)

Step 2 ：Solve for x: \(\boxed{-4}\)