Question

If \(3\begin{bmatrix} x&3\\2&1 \end{bmatrix}+4\begin{bmatrix} 1&2\\5&y \end{bmatrix}=\begin{bmatrix} 10&17\\26&11 \end{bmatrix}\)then the value of (3x+2y) is:

• A. 10
• B. 13
• C. 2
• D. 15

Answer 1

The Correct Option is A

To determine the value of \(3x + 2y\), we must solve the given equation involving matrices:
Given: \(3\begin{bmatrix} x & 3 \\ 2 & 1 \end{bmatrix} + 4\begin{bmatrix} 1 & 2 \\ 5 & y \end{bmatrix} = \begin{bmatrix} 10 & 17 \\ 26 & 11 \end{bmatrix}\).
Step 1: Expand the multiplication:
\[3\begin{bmatrix} x & 3 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} 3x & 9 \\ 6 & 3 \end{bmatrix}\]
\[4\begin{bmatrix} 1 & 2 \\ 5 & y \end{bmatrix} = \begin{bmatrix} 4 & 8 \\ 20 & 4y \end{bmatrix}\]
Step 2: Add the two matrices:
\[\begin{bmatrix} 3x & 9 \\ 6 & 3 \end{bmatrix} + \begin{bmatrix} 4 & 8 \\ 20 & 4y \end{bmatrix} = \begin{bmatrix} 3x + 4 & 17 \\ 26 & 3 + 4y \end{bmatrix}\]
Step 3: Set the results equal to the given matrix and solve for \(x\) and \(y\):
\[\begin{bmatrix} 3x + 4 & 17 \\ 26 & 3 + 4y \end{bmatrix} = \begin{bmatrix} 10 & 17 \\ 26 & 11 \end{bmatrix}\]
From the above, we deduce:
1. \(3x + 4 = 10\) — \(3x = 6\), thus \(x = 2\).
2. \(3 + 4y = 11\) — \(4y = 8\), thus \(y = 2\).
Step 4: Calculate \(3x + 2y\):
\[3x + 2y = 3(2) + 2(2) = 6 + 4 = 10\]
Therefore, the value of \(3x + 2y\) is 10.