`{:(65x - 33y = 97),(33x - 65y = 1):}`

Answer

To solve the system of equations given by:

1. \( 65x - 33y = 97 \) (Equation 1)
2. \( 33x - 65y = 1 \) (Equation 2)

we will follow a systematic approach.

### Step 1: Add the two equations
We start by adding Equation 1 and Equation 2:

\[ (65x - 33y) + (33x - 65y) = 97 + 1 \]

This simplifies to:

\[ 65x + 33x - 33y - 65y = 98 \]

Combining like terms gives:

\[ 98x - 98y = 98 \]

Dividing the entire equation by 98, we get:

\[ x - y = 1 \quad \text{(Equation 3)} \]

### Step 2: Subtract Equation 2 from Equation 1
Next, we subtract Equation 2 from Equation 1:

\[ (65x - 33y) - (33x - 65y) = 97 - 1 \]

This simplifies to:

\[ 65x - 33y - 33x + 65y = 96 \]

Combining like terms gives:

\[ 32x + 32y = 96 \]

Dividing the entire equation by 32, we get:

\[ x + y = 3 \quad \text{(Equation 4)} \]

### Step 3: Solve for \( x \) and \( y \)
Now we have two new equations:

1. \( x - y = 1 \) (Equation 3)
2. \( x + y = 3 \) (Equation 4)

We will add these two equations:

\[ (x - y) + (x + y) = 1 + 3 \]

This simplifies to:

\[ 2x = 4 \]

Dividing by 2, we find:

\[ x = 2 \]

Next, we will substitute \( x = 2 \) back into Equation 3 to find \( y \):

\[ 2 - y = 1 \]

Rearranging gives:

\[ y = 2 - 1 = 1 \]

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

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