`{:(152x - 378y = -74),(-378x + 152y = -604):}`

Answer

To solve the system of equations given by:

1) \( 152x - 378y = -74 \)
2) \( -378x + 152y = -604 \)

we will follow these steps:

### Step 1: Label the equations
Let’s label the equations for easier reference:
- Equation 1: \( 152x - 378y = -74 \)
- Equation 2: \( -378x + 152y = -604 \)

### Step 2: Add the two equations
We will add Equation 1 and Equation 2 to eliminate one of the variables.

\[ (152x - 378y) + (-378x + 152y) = -74 + (-604) \]

This simplifies to:

\[ 152x - 378x - 378y + 152y = -678 \]

Combining like terms gives us:

\[ -226x - 226y = -678 \]

### Step 3: Simplify the equation
Now, we can factor out \(-226\):

\[ -226(x + y) = -678 \]

Dividing both sides by \(-226\):

\[ x + y = \frac{678}{226} \]

Calculating the right side:

\[ x + y = 3 \]

Let’s label this as Equation 3.

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

\[ (152x - 378y) - (-378x + 152y) = -74 - (-604) \]

This simplifies to:

\[ 152x - 378y + 378x - 152y = 530 \]

Combining like terms gives us:

\[ 530x - 530y = 530 \]

### Step 5: Simplify the equation
Now, we can factor out \(530\):

\[ 530(x - y) = 530 \]

Dividing both sides by \(530\):

\[ x - y = 1 \]

Let’s label this as Equation 4.

### Step 6: Solve the system of equations
Now we have two new equations:

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

We can add these two equations:

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

This simplifies to:

\[ 2x = 4 \]

Dividing both sides by \(2\):

\[ x = 2 \]

### Step 7: Substitute to find \(y\)
Now, substitute \(x = 2\) back into Equation 3:

\[ 2 + y = 3 \]

Subtracting \(2\) from both sides gives:

\[ y = 1 \]

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

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