Question: Solve the system of linear equations for x and y
1. **State the problem:**
Solve the system of equations:
$$\begin{cases} 2x + y = 8 \\ 2x - y = 12 \end{cases}$$
2. **Formula and rules:**
We can solve this system using the method of addition (elimination) or substitution. Here, addition is convenient because adding the two equations will eliminate $y$.
3. **Add the two equations:**
$$ (2x + y) + (2x - y) = 8 + 12 $$
$$ 2x + y + 2x - y = 20 $$
$$ 4x = 20 $$
4. **Solve for $x$:**
$$ x = \frac{20}{4} $$
$$ x = 5 $$
5. **Substitute $x=5$ into the first equation to find $y$:**
$$ 2(5) + y = 8 $$
$$ 10 + y = 8 $$
$$ y = 8 - 10 $$
$$ y = -2 $$
6. **Final answer:**
$$ x = 5, \quad y = -2 $$