To solve the system of equations:
[tex]\[
\begin{cases}
2x + y = 4 \\
x + y = -2
\end{cases}
\][/tex]
we can use the method of elimination or substitution. Here, we will use the method of elimination.
### Step-by-Step Solution
1. Write down the system of equations:
[tex]\[
\begin{cases}
2x + y = 4 \quad \text{(1)} \\
x + y = -2 \quad \text{(2)}
\end{cases}
\][/tex]
2. Subtract Equation (2) from Equation (1):
[tex]\[
(2x + y) - (x + y) = 4 - (-2)
\][/tex]
Simplify the left-hand side and the right-hand side:
[tex]\[
2x + y - x - y = 4 + 2
\][/tex]
[tex]\[
x = 6
\][/tex]
Thus, we have found [tex]\( x = 6 \)[/tex].
3. Substitute [tex]\( x = 6 \)[/tex] back into one of the original equations to find [tex]\( y \)[/tex]. We will use Equation (2):
[tex]\[
x + y = -2
\][/tex]
Substitute [tex]\( x = 6 \)[/tex] into the equation:
[tex]\[
6 + y = -2
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = -2 - 6
\][/tex]
[tex]\[
y = -8
\][/tex]
4. Write the solution as an ordered pair:
The solution to the system of equations is:
[tex]\[
(x, y) = (6, -8)
\][/tex]
### Verify the Solution
To ensure our solution is correct, we can substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = -8 \)[/tex] back into the original equations.
- For Equation (1):
[tex]\[
2x + y = 4
\][/tex]
Substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = -8 \)[/tex]:
[tex]\[
2(6) + (-8) = 4
\][/tex]
[tex]\[
12 - 8 = 4
\][/tex]
[tex]\[
4 = 4 \quad \text{(True)}
\][/tex]
- For Equation (2):
[tex]\[
x + y = -2
\][/tex]
Substitute [tex]\( x = 6 \)[/tex] and [tex]\( y = -8 \)[/tex]:
[tex]\[
6 + (-8) = -2
\][/tex]
[tex]\[
-2 = -2 \quad \text{(True)}
\][/tex]
Both equations are satisfied with the solution [tex]\( (6, -8) \)[/tex]. Therefore, the solution is confirmed to be correct. The values are [tex]\( x = 6 \)[/tex] and [tex]\( y = -8 \)[/tex].
