Solve the following pair of linear equations :
`152 x -378y= -74`
`-378x+ 152y= -604`

Answer

To solve the given pair of linear equations:

1. **Equations**:
\[ 152x - 378y = -74 \quad \text{(1)} \]
\[ -378x + 152y = -604 \quad \text{(2)} \]

2. **Add the two equations**:
\[ (152x - 378y) + (-378x + 152y) = -74 + (-604) \]
Simplifying the left side:
\[ 152x - 378x - 378y + 152y = -74 - 604 \]
This results in:
\[ -226x - 226y = -678 \]
Dividing the entire equation by -226:
\[ x + y = 3 \quad \text{(3)} \]

3. **Subtract the second equation from the first**:
\[ (152x - 378y) - (-378x + 152y) = -74 - (-604) \]
Simplifying the left side:
\[ 152x - 378y + 378x - 152y = -74 + 604 \]
This results in:
\[ 530x - 530y = 530 \]
Dividing the entire equation by 530:
\[ x - y = 1 \quad \text{(4)} \]

4. **Now we have two new equations**:
\[ x + y = 3 \quad \text{(3)} \]
\[ x - y = 1 \quad \text{(4)} \]

5. **Add equations (3) and (4)**:
\[ (x + y) + (x - y) = 3 + 1 \]
This simplifies to:
\[ 2x = 4 \]
Dividing by 2 gives:
\[ x = 2 \]

6. **Substitute \(x = 2\) back into equation (3)**:
\[ 2 + y = 3 \]
Solving for \(y\):
\[ y = 3 - 2 = 1 \]

7. **Final Solution**:
The solution to the given pair of linear equations is:
\[ x = 2, \quad y = 1 \]