To determine which ordered pair is the solution to the system of equations
[tex]\[
\left\{\begin{array}{l}
x + 2y = -8 \\
y = \frac{1}{2}x - 2
\end{array}\right.,
\][/tex]
we need to verify which pair satisfies both equations.
Let's analyze each pair one by one.
### Pair [tex]\((-2, -1)\)[/tex]
1. Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -1\)[/tex] into the first equation [tex]\(x + 2y = -8\)[/tex]:
[tex]\[
-2 + 2(-1) = -2 - 2 = -4 \quad (\text{not } -8)
\][/tex]
Hence, this pair does not satisfy the first equation.
### Pair [tex]\((-2, -3)\)[/tex]
1. Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -3\)[/tex] into the first equation [tex]\(x + 2y = -8\)[/tex]:
[tex]\[
-2 + 2(-3) = -2 - 6 = -8 \quad (\text{correct})
\][/tex]
2. Substitute [tex]\(x = -2\)[/tex] and [tex]\(y = -3\)[/tex] into the second equation [tex]\(y = \frac{1}{2}x - 2\)[/tex]:
[tex]\[
-3 = \frac{1}{2}(-2) - 2 = -1 - 2 = -3 \quad (\text{correct})
\][/tex]
This pair satisfies both equations.
### Pair [tex]\((-6, -5)\)[/tex]
1. Substitute [tex]\(x = -6\)[/tex] and [tex]\(y = -5\)[/tex] into the first equation [tex]\(x + 2y = -8\)[/tex]:
[tex]\[
-6 + 2(-5) = -6 - 10 = -16 \quad (\text{not } -8)
\][/tex]
Hence, this pair does not satisfy the first equation.
### Pair [tex]\((-4, -4)\)[/tex]
1. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -4\)[/tex] into the first equation [tex]\(x + 2y = -8\)[/tex]:
[tex]\[
-4 + 2(-4) = -4 - 8 = -12 \quad (\text{not } -8)
\][/tex]
Hence, this pair does not satisfy the first equation.
Based on the above evaluations, the only ordered pair that satisfies both equations is:
[tex]\[
(-2, -3)
\][/tex]
Thus, the solution to the system of equations is [tex]\(\boxed{(-2, -3)}\)[/tex].
