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.

### Given Equation:
\[ \begin{pmatrix} 3 & -2 \\ -\frac{1}{6} & 4 \end{pmatrix} \begin{pmatrix} 2x \\ 1 \end{pmatrix} + 2 \begin{pmatrix} -4 \\ 5 \end{pmatrix} = 4 \begin{pmatrix} 2 \\ y \end{pmatrix} \]

### Step 1: Multiply the first matrix with the vector
We will first calculate the product of the matrix and the vector:
\[ \begin{pmatrix} 3 & -2 \\ -\frac{1}{6} & 4 \end{pmatrix} \begin{pmatrix} 2x \\ 1 \end{pmatrix} \]

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, we have:
\[ \begin{pmatrix} 6x - 2 \\ -\frac{x}{3} + 4 \end{pmatrix} \]

### Step 2: Add the second matrix
Now we will add \(2 \begin{pmatrix} -4 \\ 5 \end{pmatrix}\) to the result:
\[ 2 \begin{pmatrix} -4 \\ 5 \end{pmatrix} = \begin{pmatrix} -8 \\ 10 \end{pmatrix} \]

Adding the two matrices:
\[ \begin{pmatrix} 6x - 2 \\ -\frac{x}{3} + 4 \end{pmatrix} + \begin{pmatrix} -8 \\ 10 \end{pmatrix} = \begin{pmatrix} 6x - 2 - 8 \\ -\frac{x}{3} + 4 + 10 \end{pmatrix} \]
\[ = \begin{pmatrix} 6x - 10 \\ -\frac{x}{3} + 14 \end{pmatrix} \]

### Step 3: Multiply the right side
Now, let's calculate the right side:
\[ 4 \begin{pmatrix} 2 \\ y \end{pmatrix} = \begin{pmatrix} 8 \\ 4y \end{pmatrix} \]

### Step 4: Set the two sides equal
Now we set the left-hand side equal to the right-hand side:
\[ \begin{pmatrix} 6x - 10 \\ -\frac{x}{3} + 14 \end{pmatrix} = \begin{pmatrix} 8 \\ 4y \end{pmatrix} \]

### Step 5: Create equations from the matrix equality
From the first row:
\[ 6x - 10 = 8 \]
From the second row:
\[ -\frac{x}{3} + 14 = 4y \]

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

### Step 7: 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 Values
Thus, the values of \(x\) and \(y\) are:
\[ x = 3, \quad y = \frac{13}{4} \]