Solve the following pair of equations:
`43x + 67y= -24`
`67x + 43y=24`

Answer

To solve the pair of equations:

1. **Equations Given:**
\[ 43x + 67y = -24 \quad \text{(1)} \]
\[ 67x + 43y = 24 \quad \text{(2)} \]

2. **Adding the Two Equations:**
Let's add equations (1) and (2):
\[ (43x + 67y) + (67x + 43y) = -24 + 24 \]
This simplifies to:
\[ 110x + 110y = 0 \]
Dividing through by 110 gives:
\[ x + y = 0 \quad \text{(3)} \]

3. **Subtracting Equation (1) from Equation (2):**
Now, let's subtract equation (1) from equation (2):
\[ (67x + 43y) - (43x + 67y) = 24 - (-24) \]
This simplifies to:
\[ 24x - 24y = 48 \]
Dividing through by 24 gives:
\[ x - y = 2 \quad \text{(4)} \]

4. **Solving the System of Equations:**
Now we have two new equations:
\[ x + y = 0 \quad \text{(3)} \]
\[ x - y = 2 \quad \text{(4)} \]

We can solve these equations simultaneously. From equation (3), we can express \(y\) in terms of \(x\):
\[ y = -x \]

Substitute \(y = -x\) into equation (4):
\[ x - (-x) = 2 \]
This simplifies to:
\[ x + x = 2 \implies 2x = 2 \implies x = 1 \]

5. **Finding the Value of \(y\):**
Now substitute \(x = 1\) back into equation (3):
\[ 1 + y = 0 \implies y = -1 \]

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