To find which ordered pair is a solution to the system of linear equations:
[tex]$
\begin{array}{l}
x + 4y = 6 \\
y = -4x - 6
\end{array}
$[/tex]
we need to check each given pair of [tex]\((x, y)\)[/tex] in both equations to see if they satisfy both simultaneously.
### Checking [tex]\((-2, -2)\)[/tex]
1. First equation [tex]\(x + 4y = 6\)[/tex]:
[tex]\[
-2 + 4(-2) = -2 - 8 = -10 \neq 6 \quad \text{(does not satisfy)}
\][/tex]
2. Second equation [tex]\(y = -4x - 6\)[/tex]:
[tex]\[
-2 = -4(-2) - 6 = 8 - 6 = 2 \quad \text{(does not satisfy)}
\][/tex]
Since [tex]\((-2, -2)\)[/tex] does not satisfy the first equation, it is not a solution to the system.
### Checking [tex]\((-2, 2)\)[/tex]
1. First equation [tex]\(x + 4y = 6\)[/tex]:
[tex]\[
-2 + 4(2) = -2 + 8 = 6 \quad \text{(satisfies)}
\][/tex]
2. Second equation [tex]\(y = -4x - 6\)[/tex]:
[tex]\[
2 = -4(-2) - 6 = 8 - 6 = 2 \quad \text{(satisfies)}
\][/tex]
Since [tex]\((-2, 2)\)[/tex] satisfies both equations, it is a solution to the system.
### Checking [tex]\((2, 2)\)[/tex]
1. First equation [tex]\(x + 4y = 6\)[/tex]:
[tex]\[
2 + 4(2) = 2 + 8 = 10 \neq 6 \quad \text{(does not satisfy)}
\][/tex]
2. Second equation [tex]\(y = -4x - 6\)[/tex]:
[tex]\[
2 = -4(2) - 6 = -8 - 6 = -14 \quad \text{(does not satisfy)}
\][/tex]
Since [tex]\((2, 2)\)[/tex] does not satisfy the first equation, it is not a solution to the system.
### Checking [tex]\((2, -2)\)[/tex]
1. First equation [tex]\(x + 4y = 6\)[/tex]:
[tex]\[
2 + 4(-2) = 2 - 8 = -6 \neq 6 \quad \text{(does not satisfy)}
\][/tex]
2. Second equation [tex]\(y = -4x - 6\)[/tex]:
[tex]\[
-2 = -4(2) - 6 = -8 - 6 = -14 \quad \text{(does not satisfy)}
\][/tex]
Since [tex]\((2, -2)\)[/tex] does not satisfy the first equation, it is not a solution to the system.
### Conclusion
The only ordered pair that satisfies both equations in the system is [tex]\((-2, 2)\)[/tex]. Therefore, the solution to the system of linear equations is:
[tex]\[
\boxed{2}
\][/tex]
