If `[(x+y,y),(2x,x-y)][(2),(-1)]=[3/2]` then x.y is equal to :

Options

A. `-5`
B. 5
C. 4
D. 6

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) given the equation:

\[ \begin{pmatrix} x+y & y \\ 2x & x-y \end{pmatrix} \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \]

### Step 1: Matrix Multiplication
We start by performing the multiplication of the two matrices on the left side.

The first row of the resulting matrix is calculated as follows:
\[ (x+y) \cdot 2 + y \cdot (-1) = 2(x+y) - y = 2x + 2y - y = 2x + y \]

The second row of the resulting matrix is calculated as follows:
\[ 2x \cdot 2 + (x-y) \cdot (-1) = 4x - (x - y) = 4x - x + y = 3x + y \]

Thus, the resulting matrix after multiplication is:
\[ \begin{pmatrix} 2x + y \\ 3x + y \end{pmatrix} \]

### Step 2: Set Up the Equations
Now we can set this equal to the matrix on the right side:
\[ \begin{pmatrix} 2x + y \\ 3x + y \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix} \]

From this, we can form two equations:
1. \( 2x + y = 3 \) (Equation 1)
2. \( 3x + y = 2 \) (Equation 2)

### Step 3: Solve the Equations
Next, we will solve these two equations simultaneously. We can eliminate \( y \) by subtracting Equation 2 from Equation 1:
\[ (2x + y) - (3x + y) = 3 - 2 \]
This simplifies to:
\[ 2x + y - 3x - y = 1 \\ -x = 1 \\ x = -1 \]

### Step 4: Substitute to Find \( y \)
Now that we have \( x = -1 \), we can substitute this value back into either of the original equations to find \( y \). We will use Equation 1:
\[ 2(-1) + y = 3 \\ -2 + y = 3 \\ y = 3 + 2 \\ y = 5 \]

### Step 5: Calculate \( x \cdot y \)
Now that we have both \( x \) and \( y \):
\[ x = -1, \quad y = 5 \]
We can find the product \( x \cdot y \):
\[ x \cdot y = -1 \cdot 5 = -5 \]

### Final Answer
Thus, the value of \( x \cdot y \) is:
\[ \boxed{-5} \]