Solve: `{:(2x - 5y + 4 = 0),(2x + y - 8 = 0):}`

Answer

To solve the system of equations:

1. **Equations Given:**
\[ \begin{align*} (1) & \quad 2x - 5y + 4 = 0 \\ (2) & \quad 2x + y - 8 = 0 \end{align*} \]

2. **Rearranging the Equations:**
Let's rearrange both equations to express them in a more standard form:
\[ \begin{align*} (1) & \quad 2x - 5y = -4 \\ (2) & \quad 2x + y = 8 \end{align*} \]

3. **Using the Elimination Method:**
Since the coefficients of \(x\) are the same in both equations, we can use the elimination method. We will subtract equation (2) from equation (1).

Subtracting (2) from (1):
\[ (2x - 5y) - (2x + y) = -4 - 8 \]

Simplifying:
\[ 2x - 5y - 2x - y = -4 - 8 \]
\[ -6y = -12 \]

4. **Solving for \(y\):**
Dividing both sides by -6:
\[ y = \frac{-12}{-6} = 2 \]

5. **Substituting \(y\) back into one of the original equations:**
We can substitute \(y = 2\) back into equation (2):
\[ 2x + y = 8 \]
\[ 2x + 2 = 8 \]

6. **Solving for \(x\):**
Subtracting 2 from both sides:
\[ 2x = 8 - 2 \]
\[ 2x = 6 \]
Dividing both sides by 2:
\[ x = \frac{6}{2} = 3 \]

7. **Final Solution:**
The solution to the system of equations is:
\[ x = 3, \quad y = 2 \]

---