`{:(217x + 131y = 913),(131x + 217y = 827):}`

Answer

To solve the system of equations given by:

1) \( 217x + 131y = 913 \)
2) \( 131x + 217y = 827 \)

we will follow these steps:

### Step 1: Label the equations
Let’s label the equations for reference:
- Equation (1): \( 217x + 131y = 913 \)
- Equation (2): \( 131x + 217y = 827 \)

### Step 2: Add the equations
Now, we will add both equations (1) and (2):

\[ (217x + 131y) + (131x + 217y) = 913 + 827 \]

This simplifies to:

\[ 348x + 348y = 1740 \]

### Step 3: Simplify the equation
Now, we can divide the entire equation by 348 to simplify it:

\[ x + y = 5 \]

Let’s label this as Equation (3).

### Step 4: Subtract the equations
Next, we will subtract Equation (2) from Equation (1):

\[ (217x + 131y) - (131x + 217y) = 913 - 827 \]

This simplifies to:

\[ 86x - 86y = 86 \]

### Step 5: Simplify the equation
Now, we can divide the entire equation by 86:

\[ x - y = 1 \]

Let’s label this as Equation (4).

### Step 6: Solve the system of equations
Now we have two new equations:
- Equation (3): \( x + y = 5 \)
- Equation (4): \( x - y = 1 \)

We can add these two equations:

\[ (x + y) + (x - y) = 5 + 1 \]

This simplifies to:

\[ 2x = 6 \]

### Step 7: Find the value of \( x \)
Now, divide both sides by 2:

\[ x = 3 \]

### Step 8: Substitute to find \( y \)
Now that we have \( x \), we can substitute it back into Equation (3) to find \( y \):

\[ 3 + y = 5 \]

Subtracting 3 from both sides gives:

\[ y = 2 \]

### Final Answer
Thus, the solution to the system of equations is:

\[ x = 3, \quad y = 2 \]

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