Question

The value of 2y-3x, if
\(2\begin {bmatrix}x &5\\ 7&y-3\end{bmatrix}+\begin{bmatrix}3&-4\\ 1&2\end{bmatrix}=\begin{bmatrix}7&6 \\15&14\end{bmatrix}\) is:

• A. 12
• B. -10
• C. 6
• D. -5

Answer 1

The Correct Option is A

To solve for the value of \(2y - 3x\) given the matrix equation: \(2\begin {bmatrix}x &5\\ 7&y-3\end{bmatrix}+\begin{bmatrix}3&-4\\ 1&2\end{bmatrix}=\begin{bmatrix}7&6 \\15&14\end{bmatrix}\), we follow these steps:

1. Distribute the 2 into the first matrix:

\(2\begin{bmatrix}x & 5 \\ 7 & y-3\end{bmatrix} = \begin{bmatrix}2x & 10 \\ 14 & 2(y-3)\end{bmatrix} = \begin{bmatrix}2x & 10 \\ 14 & 2y-6\end{bmatrix}\)

2. Add the second matrix:

\(\begin{bmatrix}2x & 10 \\ 14 & 2y-6\end{bmatrix} + \begin{bmatrix}3 & -4 \\ 1 & 2\end{bmatrix} = \begin{bmatrix}2x+3 & 10-4 \\ 14+1 & 2y-6+2\end{bmatrix} = \begin{bmatrix}2x+3 & 6 \\ 15 & 2y-4\end{bmatrix}\)

3. Set this equal to the given matrix:

\(\begin{bmatrix}2x+3 & 6 \\ 15 & 2y-4\end{bmatrix} = \begin{bmatrix}7 & 6 \\ 15 & 14\end{bmatrix}\)

4. Equate corresponding elements:

- For the first row, \(2x+3 = 7\)

- For the second row, \(2y-4 = 14\)

5. Solve these equations:

- \(2x+3 = 7 \Rightarrow 2x = 4 \Rightarrow x = 2\)

- \(2y-4 = 14 \Rightarrow 2y = 18 \Rightarrow y = 9\)

6. Find \(2y - 3x\):

- \(2y - 3x = 2(9) - 3(2) = 18 - 6 = 12\)

Therefore, the value of \(2y - 3x\) is 12.