If `[(3,-2),(-1//6,4)][(2x),(1)]+2[(-4),(5)]=4[(2),(y)]` then find the values of x and y.

Options

A. `(13)/4,1`
B. 0,3
C. `3,(13)/4`
D. 3,0

Answer

To solve the equation given in the question, we will break it down step by step.

### Step 1: Write down the equation
The equation is given as:
\[ \begin{bmatrix} 3 & -2 \\ -\frac{1}{6} & 4 \end{bmatrix} \begin{bmatrix} 2x \\ 1 \end{bmatrix} + 2 \begin{bmatrix} -4 \\ 5 \end{bmatrix} = 4 \begin{bmatrix} 2 \\ y \end{bmatrix} \]

### Step 2: Multiply the matrices
First, we will multiply the 2x2 matrix with the 2x1 matrix:
\[ \begin{bmatrix} 3 & -2 \\ -\frac{1}{6} & 4 \end{bmatrix} \begin{bmatrix} 2x \\ 1 \end{bmatrix} \]
Calculating the first row:
\[ 3(2x) + (-2)(1) = 6x - 2 \]
Calculating the second row:
\[ -\frac{1}{6}(2x) + 4(1) = -\frac{2x}{6} + 4 = -\frac{x}{3} + 4 \]
So, the result of the multiplication is:
\[ \begin{bmatrix} 6x - 2 \\ -\frac{x}{3} + 4 \end{bmatrix} \]

### Step 3: Multiply the scalar with the second matrix
Next, we will calculate \(2 \begin{bmatrix} -4 \\ 5 \end{bmatrix}\):
\[ 2 \begin{bmatrix} -4 \\ 5 \end{bmatrix} = \begin{bmatrix} -8 \\ 10 \end{bmatrix} \]

### Step 4: Add the results
Now, we add the two results:
\[ \begin{bmatrix} 6x - 2 \\ -\frac{x}{3} + 4 \end{bmatrix} + \begin{bmatrix} -8 \\ 10 \end{bmatrix} = \begin{bmatrix} (6x - 2 - 8) \\ (-\frac{x}{3} + 4 + 10) \end{bmatrix} = \begin{bmatrix} 6x - 10 \\ -\frac{x}{3} + 14 \end{bmatrix} \]

### Step 5: Calculate the right side
Now calculate the right side:
\[ 4 \begin{bmatrix} 2 \\ y \end{bmatrix} = \begin{bmatrix} 8 \\ 4y \end{bmatrix} \]

### Step 6: Set the matrices equal
Now we set the two results equal to each other:
\[ \begin{bmatrix} 6x - 10 \\ -\frac{x}{3} + 14 \end{bmatrix} = \begin{bmatrix} 8 \\ 4y \end{bmatrix} \]

### Step 7: Create equations
From the equality of the matrices, we can create two equations:
1. \(6x - 10 = 8\)
2. \(-\frac{x}{3} + 14 = 4y\)

### Step 8: Solve for \(x\)
From the first equation:
\[ 6x - 10 = 8 \implies 6x = 18 \implies x = 3 \]

### Step 9: Substitute \(x\) into the second equation
Now substitute \(x = 3\) into the second equation:
\[ -\frac{3}{3} + 14 = 4y \implies -1 + 14 = 4y \implies 13 = 4y \implies y = \frac{13}{4} \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:
\[ x = 3, \quad y = \frac{13}{4} \]