If \(a = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\) and \(b = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\), find a vector c such that \(4a + 3c = b\).

If \(a = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\) and \(b = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\), find a vector c such that \(4a + 3c = b\).

Answer Details

To solve this problem, we want to find a vector c that satisfies the equation 4a + 3c = b. We can rearrange this equation to get: 3c = b - 4a Now we just need to solve for c. We can do this by dividing both sides by 3: c = (b - 4a) / 3 Substituting the values of a and b given in the problem, we get: c = (\begin{pmatrix} -3 \\ 5 \end{pmatrix} - 4\begin{pmatrix} 3 \\ 2 \end{pmatrix}) / 3 c = (\begin{pmatrix} -3 \\ 5 \end{pmatrix} - \begin{pmatrix} 12 \\ 8 \end{pmatrix}) / 3 c = \begin{pmatrix} -15/3 \\ -3/3 \end{pmatrix} c = \begin{pmatrix} -5 \\ -1 \end{pmatrix} Therefore, the answer is option B, \(\begin{pmatrix} -5 \\ -1 \end{pmatrix}\).