To solve this problem, we just need to perform the matrix multiplication on the left-hand side and equate it with the right-hand side, and then solve for the unknown variable x.
\(\begin{pmatrix} 3 & 2 \\ 7 & x \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 12 \\ 29 \end{pmatrix}\)
Performing the matrix multiplication on the left-hand side gives:
\(\begin{pmatrix} 3(2) + 2(3) \\ 7(2) + x(3) \end{pmatrix} = \begin{pmatrix} 12 \\ 29 \end{pmatrix}\)
Simplifying the left-hand side gives:
\(\begin{pmatrix} 12 \\ 14 + 3x \end{pmatrix} = \begin{pmatrix} 12 \\ 29 \end{pmatrix}\)
We can see that the first element of both matrices is already equal to 12, so we only need to equate the second elements:
\(14 + 3x = 29\)
Solving for x gives:
\(3x = 15\)
\(x = 5\)
Therefore, the answer is x = 5.