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

Answer

To solve the system of equations given by:

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

we will follow these steps:

### Step 1: Rewrite the equations
We can keep the equations as they are for now.

### Step 2: Add the two equations
We will add Equation 1 and Equation 2 together:

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

This simplifies to:

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

Combining like terms gives:

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

### Step 3: Simplify the equation
We can divide the entire equation by -226 to simplify:

\[ x + y = 3 \]

This is our new Equation 3.

### Step 4: Subtract Equation 2 from Equation 1
Now, we will subtract Equation 2 from Equation 1:

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

This simplifies to:

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

Combining like terms gives:

\[ 530x - 530y = 530 \]

### Step 5: Simplify the equation
We can divide the entire equation by 530:

\[ x - y = 1 \]

This is our new Equation 4.

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

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

We can add these two equations together:

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

This simplifies to:

\[ 2x = 4 \]

Dividing by 2 gives:

\[ x = 2 \]

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

\[ 2 + y = 3 \]

Solving for \( y \) gives:

\[ y = 1 \]

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

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

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