Solve the given pair of equations:
` 23 x - 29 y = 98` ,
` 29 x - 23 y = 110`

Answer

To solve the given pair of equations:

1. **Equations:**
\[ 23x - 29y = 98 \quad \text{(1)} \]
\[ 29x - 23y = 110 \quad \text{(2)} \]

2. **Add the equations:**
To eliminate one of the variables, we can add both equations. First, we rewrite them:
\[ (23x - 29y) + (29x - 23y) = 98 + 110 \]
Combining like terms:
\[ 52x - 52y = 208 \]
We can factor out 52:
\[ 52(x - y) = 208 \]
Dividing both sides by 52:
\[ x - y = 4 \quad \text{(3)} \]

3. **Subtract the equations:**
Now, we subtract equation (2) from equation (1):
\[ (23x - 29y) - (29x - 23y) = 98 - 110 \]
Simplifying this gives:
\[ 23x - 29y - 29x + 23y = -12 \]
Combining like terms:
\[ -6x - 6y = -12 \]
Dividing through by -6:
\[ x + y = 2 \quad \text{(4)} \]

4. **Solve the new system of equations:**
Now we have a simpler system of equations:
\[ x - y = 4 \quad \text{(3)} \]
\[ x + y = 2 \quad \text{(4)} \]
We can add equations (3) and (4):
\[ (x - y) + (x + y) = 4 + 2 \]
This simplifies to:
\[ 2x = 6 \]
Dividing by 2:
\[ x = 3 \]

5. **Substitute to find y:**
Now we substitute \( x = 3 \) back into equation (4):
\[ 3 + y = 2 \]
Solving for \( y \):
\[ y = 2 - 3 = -1 \]

6. **Final solution:**
The solution to the system of equations is:
\[ x = 3, \quad y = -1 \]