`{:(3x + 2y = 4),(2x - 3y = 7):}`

Answer

To solve the system of equations given by:

1. \( 3x + 2y = 4 \) (Equation 1)
2. \( 2x - 3y = 7 \) (Equation 2)

we will use the elimination method. Here are the steps:

### Step 1: Make the coefficients of \( y \) the same
To eliminate \( y \), we need to make the coefficients of \( y \) in both equations the same. The coefficients are 2 in Equation 1 and -3 in Equation 2. We can achieve this by multiplying Equation 1 by 3 and Equation 2 by 2.

- Multiply Equation 1 by 3:
\[ 3(3x + 2y) = 3(4) \implies 9x + 6y = 12 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 2:
\[ 2(2x - 3y) = 2(7) \implies 4x - 6y = 14 \quad \text{(Equation 4)} \]

### Step 2: Add the new equations
Now we will add Equation 3 and Equation 4 to eliminate \( y \):

\[ (9x + 6y) + (4x - 6y) = 12 + 14 \]

This simplifies to:
\[ 9x + 4x + 6y - 6y = 12 + 14 \]
\[ 13x = 26 \]

### Step 3: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{26}{13} = 2 \]

### Step 4: Substitute \( x \) back into one of the original equations
Now that we have \( x = 2 \), we can substitute this value back into Equation 1 to find \( y \):

\[ 3(2) + 2y = 4 \]
\[ 6 + 2y = 4 \]
\[ 2y = 4 - 6 \]
\[ 2y = -2 \]
\[ y = \frac{-2}{2} = -1 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 2, \quad y = -1 \]