`{:(3x - 2y = 4),(5x - 2y = 0):}`

Answer

To solve the given system of linear equations:

1. **Equations Given:**
- Equation 1: \(3x - 2y = 4\)
- Equation 2: \(5x - 2y = 0\)

2. **Using the Elimination Method:**
Since both equations have the same coefficient for \(y\) (-2), we can eliminate \(y\) by subtracting one equation from the other.

3. **Subtract Equation 2 from Equation 1:**
\[ (3x - 2y) - (5x - 2y) = 4 - 0 \]
Simplifying this gives:
\[ 3x - 5x - 2y + 2y = 4 \]
This simplifies to:
\[ -2x = 4 \]

4. **Solving for \(x\):**
Divide both sides by -2:
\[ x = \frac{4}{-2} = -2 \]

5. **Substituting \(x\) back to find \(y\):**
Now that we have \(x = -2\), we can substitute this value into either of the original equations. We will use Equation 2:
\[ 5x - 2y = 0 \]
Substituting \(x = -2\):
\[ 5(-2) - 2y = 0 \]
This simplifies to:
\[ -10 - 2y = 0 \]

6. **Solving for \(y\):**
Rearranging gives:
\[ -2y = 10 \]
Dividing both sides by -2:
\[ y = \frac{10}{-2} = -5 \]

7. **Final Solution:**
The solution to the system of equations is:
\[ x = -2, \quad y = -5 \]