If \(\begin{pmatrix} 3 & 2 \\ 7 & x \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 12 \\ 29 \end{pmatrix} \), find x.

Answer Details

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.