To identify the ordered pair that is the solution to the given system of equations, we need to verify which of the provided pairs satisfies both equations simultaneously. The system of equations is:
[tex]\[
\begin{cases}
x = \frac{1}{2} y + 5 \\
2x + 3y = -14
\end{cases}
\][/tex]
We will test each ordered pair [tex]\((x, y)\)[/tex] in turn.
### Pair 1: [tex]\((-1, -12)\)[/tex]
1. Substitute [tex]\((-1, -12)\)[/tex] into [tex]\( x = \frac{1}{2} y + 5 \)[/tex]:
[tex]\[
-1 = \frac{1}{2}(-12) + 5 \\
-1 = -6 + 5 \\
-1 = -1 \quad \text{(This is true)}
\][/tex]
2. Substitute [tex]\((-1, -12)\)[/tex] into [tex]\( 2x + 3y = -14 \)[/tex]:
[tex]\[
2(-1) + 3(-12) = -14 \\
-2 - 36 = -14 \\
-38 \neq -14 \quad \text{(This is false)}
\][/tex]
Since the second equation is not satisfied, [tex]\((-1, -12)\)[/tex] is not a solution.
### Pair 2: [tex]\((2, -6)\)[/tex]
1. Substitute [tex]\((2, -6)\)[/tex] into [tex]\( x = \frac{1}{2} y + 5 \)[/tex]:
[tex]\[
2 = \frac{1}{2}(-6) + 5 \\
2 = -3 + 5 \\
2 = 2 \quad \text{(This is true)}
\][/tex]
2. Substitute [tex]\((2, -6)\)[/tex] into [tex]\( 2x + 3y = -14 \)[/tex]:
[tex]\[
2(2) + 3(-6) = -14 \\
4 - 18 = -14 \\
-14 = -14 \quad \text{(This is true)}
\][/tex]
Since both equations are satisfied, [tex]\((2, -6)\)[/tex] is a solution.
### Pair 3: [tex]\((18, 14)\)[/tex]
1. Substitute [tex]\((18, 14)\)[/tex] into [tex]\( x = \frac{1}{2} y + 5 \)[/tex]:
[tex]\[
18 = \frac{1}{2}(14) + 5 \\
18 = 7 + 5 \\
18 = 12 \quad \text{(This is false)}
\][/tex]
Since the first equation is not satisfied, we do not need to check the second equation. [tex]\((18, 14)\)[/tex] is not a solution.
### Pair 4: [tex]\((8, 9)\)[/tex]
1. Substitute [tex]\((8, 9)\)[/tex] into [tex]\( x = \frac{1}{2} y + 5 \)[/tex]:
[tex]\[
8 = \frac{1}{2}(9) + 5 \\
8 = 4.5 + 5 \\
8 = 9.5 \quad \text{(This is false)}
\][/tex]
Since the first equation is not satisfied, we do not need to check the second equation. [tex]\((8, 9)\)[/tex] is not a solution.
### Conclusion
The ordered pair that satisfies both equations is [tex]\((2, -6)\)[/tex]. Thus, the solution to the system of equations is [tex]\((2, -6)\)[/tex].
