`{:(2x - 3y = 17),(4x + y = 13):}`

Answer

To solve the system of equations given by:

1. \( 2x - 3y = 17 \) (Equation 1)
2. \( 4x + y = 13 \) (Equation 2)

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

### Step 1: Prepare the equations for elimination
We want to eliminate one of the variables by making the coefficients of \( y \) the same in both equations. To do this, we can multiply Equation 2 by 3.

\[ 3(4x + y) = 3(13) \]

This gives us:

\[ 12x + 3y = 39 \quad \text{(Equation 3)} \]

### Step 2: Write down the modified equations
Now we have:

1. \( 2x - 3y = 17 \) (Equation 1)
2. \( 12x + 3y = 39 \) (Equation 3)

### Step 3: Add the equations
Next, we will add Equation 1 and Equation 3 together to eliminate \( y \):

\[ (2x - 3y) + (12x + 3y) = 17 + 39 \]

This simplifies to:

\[ 2x + 12x - 3y + 3y = 56 \]

\[ 14x = 56 \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):

\[ x = \frac{56}{14} = 4 \]

### Step 5: Substitute \( x \) back into one of the original equations
Now that we have \( x = 4 \), we can substitute this value back into either of the original equations to find \( y \). We will use Equation 1:

\[ 2(4) - 3y = 17 \]

This simplifies to:

\[ 8 - 3y = 17 \]

### Step 6: Solve for \( y \)
Now, isolate \( y \):

\[ -3y = 17 - 8 \]

\[ -3y = 9 \]

\[ y = \frac{9}{-3} = -3 \]

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

\[ x = 4, \quad y = -3 \]