Find the value of 'x' and 'y' if :
`2[(x,7),(9,y-5)]+[(6,-7),(4,5)]=[(10,7),(22,15)]`

Answer

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

### Step 1: Write down the equation
We start with the equation:
\[ 2 \begin{pmatrix} x & 7 \\ 9 & y-5 \end{pmatrix} + \begin{pmatrix} 6 & -7 \\ 4 & 5 \end{pmatrix} = \begin{pmatrix} 10 & 7 \\ 22 & 15 \end{pmatrix} \]

### Step 2: Multiply the first matrix by 2
We will multiply the first matrix by 2:
\[ 2 \begin{pmatrix} x & 7 \\ 9 & y-5 \end{pmatrix} = \begin{pmatrix} 2x & 14 \\ 18 & 2(y-5) \end{pmatrix} = \begin{pmatrix} 2x & 14 \\ 18 & 2y - 10 \end{pmatrix} \]

### Step 3: Add the second matrix
Now we will add the second matrix to the result:
\[ \begin{pmatrix} 2x & 14 \\ 18 & 2y - 10 \end{pmatrix} + \begin{pmatrix} 6 & -7 \\ 4 & 5 \end{pmatrix} = \begin{pmatrix} 2x + 6 & 14 - 7 \\ 18 + 4 & 2y - 10 + 5 \end{pmatrix} \]
This simplifies to:
\[ \begin{pmatrix} 2x + 6 & 7 \\ 22 & 2y - 5 \end{pmatrix} \]

### Step 4: Set up equations
Now we set this equal to the matrix on the right side:
\[ \begin{pmatrix} 2x + 6 & 7 \\ 22 & 2y - 5 \end{pmatrix} = \begin{pmatrix} 10 & 7 \\ 22 & 15 \end{pmatrix} \]

From this, we can create the following equations by comparing corresponding elements:
1. \( 2x + 6 = 10 \)
2. \( 2y - 5 = 15 \)

### Step 5: Solve for x
From the first equation:
\[ 2x + 6 = 10 \]
Subtract 6 from both sides:
\[ 2x = 4 \]
Now divide by 2:
\[ x = 2 \]

### Step 6: Solve for y
From the second equation:
\[ 2y - 5 = 15 \]
Add 5 to both sides:
\[ 2y = 20 \]
Now divide by 2:
\[ y = 10 \]

### Final Answer
Thus, the values of \( x \) and \( y \) are:
\[ x = 2 \]
\[ y = 10 \]

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