Find x and y, if : `[{:(,x,3x),(,y,4y):}] [{:(,2),(,1):}]=[{:(,5),(,12):}]`

Answer

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

\[ \begin{pmatrix} x & 3x \\ y & 4y \end{pmatrix} \begin{pmatrix} 2 & 1 \\ \end{pmatrix} = \begin{pmatrix} 5 & 12 \end{pmatrix} \]

### Step 1: Multiply the matrices

We start by multiplying the first matrix by the second matrix. The multiplication of the first row of the first matrix by the first column of the second matrix gives us:

\[ x \cdot 2 + 3x \cdot 1 = 2x + 3x = 5x \]

Similarly, for the second row of the first matrix:

\[ y \cdot 2 + 4y \cdot 1 = 2y + 4y = 6y \]

### Step 2: Set up the equations

From the multiplication, we have two equations based on the equality of the matrices:

1. \( 5x = 5 \)
2. \( 6y = 12 \)

### Step 3: Solve for \( x \)

Now, we can solve for \( x \) from the first equation:

\[ 5x = 5 \]
Dividing both sides by 5:

\[ x = 1 \]

### Step 4: Solve for \( y \)

Next, we solve for \( y \) from the second equation:

\[ 6y = 12 \]
Dividing both sides by 6:

\[ y = 2 \]

### Final Answer

Thus, the values of \( x \) and \( y \) are:

\[ x = 1, \quad y = 2 \]

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