Find x and y if
`[{:(,3,-2),(,-1,4):}] [{:(,2x),(,1):}] +2 [{:(,-4),(,5):}] =4[{:(,2),(,y):}]`

Answer

To solve the equation given in the question, we need to manipulate the matrices step by step. Let's break it down:

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

### Step 1: Multiply the first matrix with the vector
We start by multiplying the first matrix with the vector:
\[ \begin{pmatrix} 3 & -2 \\ -1 & 4 \end{pmatrix} \begin{pmatrix} 2x \\ 1 \end{pmatrix} \]

Calculating the multiplication:
- For the first row: \(3(2x) + (-2)(1) = 6x - 2\)
- For the second row: \(-1(2x) + 4(1) = -2x + 4\)

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

### Step 2: Multiply the scalar with the second matrix
Next, we calculate:
\[ 2 \begin{pmatrix} -4 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \cdot -4 \\ 2 \cdot 5 \end{pmatrix} = \begin{pmatrix} -8 \\ 10 \end{pmatrix} \]

### Step 3: Add the results from Step 1 and Step 2
Now, we add the two results:
\[ \begin{pmatrix} 6x - 2 \\ -2x + 4 \end{pmatrix} + \begin{pmatrix} -8 \\ 10 \end{pmatrix} = \begin{pmatrix} (6x - 2) + (-8) \\ (-2x + 4) + 10 \end{pmatrix} \]

Calculating the components:
- First component: \(6x - 2 - 8 = 6x - 10\)
- Second component: \(-2x + 4 + 10 = -2x + 14\)

So, we have:
\[ \begin{pmatrix} 6x - 10 \\ -2x + 14 \end{pmatrix} \]

### Step 4: Multiply the right side of the equation
Now, we calculate the right side:
\[ 4 \begin{pmatrix} 2 \\ y \end{pmatrix} = \begin{pmatrix} 4 \cdot 2 \\ 4 \cdot y \end{pmatrix} = \begin{pmatrix} 8 \\ 4y \end{pmatrix} \]

### Step 5: Set the components equal to each other
Now, we set the two matrices equal to each other:
\[ \begin{pmatrix} 6x - 10 \\ -2x + 14 \end{pmatrix} = \begin{pmatrix} 8 \\ 4y \end{pmatrix} \]

This gives us two equations:
1. \(6x - 10 = 8\)
2. \(-2x + 14 = 4y\)

### Step 6: Solve the first equation for x
From the first equation:
\[ 6x - 10 = 8 \]
Add 10 to both sides:
\[ 6x = 18 \]
Now, divide by 6:
\[ x = 3 \]

### Step 7: Substitute x into the second equation to find y
Now substitute \(x = 3\) into the second equation:
\[ -2(3) + 14 = 4y \]
This simplifies to:
\[ -6 + 14 = 4y \]
\[ 8 = 4y \]
Now, divide by 4:
\[ y = 2 \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:
\[ x = 3, \quad y = 2 \]