Find x, y if `[(-2,0),(3,1)][(-1),(2x)]+[(-2),(1)]=2[(y),(3)]`

Answer

To solve the equation given in the question, we will follow these steps:

### Step 1: Write down the matrices
We have the equation:
\[ \begin{pmatrix} -2 & 3 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} -1 \\ 2x \end{pmatrix} + \begin{pmatrix} -2 \\ 1 \end{pmatrix} = 2 \begin{pmatrix} y \\ 3 \end{pmatrix} \]

### Step 2: Perform the matrix multiplication
First, we will multiply the first matrix with the second matrix:
\[ \begin{pmatrix} -2 & 3 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} -1 \\ 2x \end{pmatrix} = \begin{pmatrix} (-2)(-1) + (3)(2x) \\ (0)(-1) + (1)(2x) \end{pmatrix} = \begin{pmatrix} 2 + 6x \\ 2x \end{pmatrix} \]

### Step 3: Add the second matrix
Now we add the result to the second matrix:
\[ \begin{pmatrix} 2 + 6x \\ 2x \end{pmatrix} + \begin{pmatrix} -2 \\ 1 \end{pmatrix} = \begin{pmatrix} (2 + 6x - 2) \\ (2x + 1) \end{pmatrix} = \begin{pmatrix} 6x \\ 2x + 1 \end{pmatrix} \]

### Step 4: Set the equation equal to the right-hand side
Now we set this equal to the right-hand side:
\[ \begin{pmatrix} 6x \\ 2x + 1 \end{pmatrix} = \begin{pmatrix} 2y \\ 6 \end{pmatrix} \]

### Step 5: Create equations from the matrix equality
From the equality of the matrices, we can create two equations:
1. \( 6x = 2y \)
2. \( 2x + 1 = 6 \)

### Step 6: Solve the second equation for x
From the second equation:
\[ 2x + 1 = 6 \]
Subtract 1 from both sides:
\[ 2x = 5 \]
Now divide by 2:
\[ x = \frac{5}{2} = 2.5 \]

### Step 7: Substitute x into the first equation to find y
Now we substitute \( x = 2.5 \) into the first equation:
\[ 6(2.5) = 2y \]
This gives:
\[ 15 = 2y \]
Now divide by 2:
\[ y = \frac{15}{2} = 7.5 \]

### Final Solution
Thus, the values of \( x \) and \( y \) are:
\[ x = 2.5, \quad y = 7.5 \]