To determine which ordered pair is the solution to the given system of equations:
[tex]\[
\begin{cases}
x = \frac{1}{2} y + 5 \\
2x + 3y = -14
\end{cases}
\][/tex]
we need to test each pair separately to check if it satisfies both equations.
### Checking the pair [tex]\((18, 14)\)[/tex]:
1. Substitute [tex]\( x = 18 \)[/tex] and [tex]\( y = 14 \)[/tex] into the first equation:
[tex]\[
x = \frac{1}{2}y + 5
\][/tex]
[tex]\[
18 = \frac{1}{2}(14) + 5
\][/tex]
[tex]\[
18 = 7 + 5
\][/tex]
[tex]\[
18 = 12 \quad (\text{False})
\][/tex]
Since this does not satisfy the first equation, the pair [tex]\((18, 14)\)[/tex] is not a solution.
### Checking the pair [tex]\((8, 9)\)[/tex]:
1. Substitute [tex]\( x = 8 \)[/tex] and [tex]\( y = 9 \)[/tex] into the first equation:
[tex]\[
x = \frac{1}{2}y + 5
\][/tex]
[tex]\[
8 = \frac{1}{2}(9) + 5
\][/tex]
[tex]\[
8 = 4.5 + 5
\][/tex]
[tex]\[
8 = 9.5 \quad (\text{False})
\][/tex]
Since this does not satisfy the first equation, the pair [tex]\((8, 9)\)[/tex] is not a solution.
### Checking the pair [tex]\((-1, -12)\)[/tex]:
1. Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -12 \)[/tex] into the first equation:
[tex]\[
x = \frac{1}{2}y + 5
\][/tex]
[tex]\[
-1 = \frac{1}{2}(-12) + 5
\][/tex]
[tex]\[
-1 = -6 + 5
\][/tex]
[tex]\[
-1 = -1 \quad (\text{True})
\][/tex]
2. Now, substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -12 \)[/tex] into the second equation:
[tex]\[
2x + 3y = -14
\][/tex]
[tex]\[
2(-1) + 3(-12) = -14
\][/tex]
[tex]\[
-2 - 36 = -14
\][/tex]
[tex]\[
-38 \neq -14 \quad (\text{False})
\][/tex]
Since this does not satisfy the second equation, the pair [tex]\((-1, -12)\)[/tex] is not a solution.
### Checking the pair [tex]\((2, -6)\)[/tex]:
1. Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = -6 \)[/tex] into the first equation:
[tex]\[
x = \frac{1}{2}y + 5
\][/tex]
[tex]\[
2 = \frac{1}{2}(-6) + 5
\][/tex]
[tex]\[
2 = -3 + 5
\][/tex]
[tex]\[
2 = 2 \quad (\text{True})
\][/tex]
2. Now, substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = -6 \)[/tex] into the second equation:
[tex]\[
2x + 3y = -14
\][/tex]
[tex]\[
2(2) + 3(-6) = -14
\][/tex]
[tex]\[
4 - 18 = -14
\][/tex]
[tex]\[
-14 = -14 \quad (\text{True})
\][/tex]
Since this satisfies both equations, the pair [tex]\((2, -6)\)[/tex] is the solution.
Therefore, the ordered pair that is the solution to the system of equations is [tex]\((2, -6)\)[/tex].
