To determine which system of equations has the solution [tex]\((-1, 5)\)[/tex], we need to check each system and substitute [tex]\(x = -1\)[/tex] and [tex]\(y = 5\)[/tex] into the equations to see if both equations in a given system are satisfied.
### System 1:
[tex]\[
\begin{cases}
y = 3x + 8 \\
y = -x + 4
\end{cases}
\][/tex]
1. Substitute [tex]\(x = -1\)[/tex] into [tex]\( y = 3x + 8 \)[/tex]:
[tex]\[
y = 3(-1) + 8 = -3 + 8 = 5
\][/tex]
2. Substitute [tex]\(x = -1\)[/tex] into [tex]\( y = -x + 4 \)[/tex]:
[tex]\[
y = -(-1) + 4 = 1 + 4 = 5
\][/tex]
Both equations are satisfied, so System 1 has the solution [tex]\((-1, 5)\)[/tex].
### System 2:
[tex]\[
\begin{cases}
y = 3x + 8 \\
y = -x - 4
\end{cases}
\][/tex]
1. Substitute [tex]\(x = -1\)[/tex] into [tex]\( y = 3x + 8 \)[/tex]:
[tex]\[
y = 3(-1) + 8 = -3 + 8 = 5
\][/tex]
2. Substitute [tex]\(x = -1\)[/tex] into [tex]\( y = -x - 4 \)[/tex]:
[tex]\[
y = -(-1) - 4 = 1 - 4 = -3
\][/tex]
The second equation is not satisfied, so System 2 does not have the solution [tex]\((-1, 5)\)[/tex].
### System 3:
[tex]\[
\begin{cases}
y = 3x - 8 \\
y = -x + 4
\end{cases}
\][/tex]
1. Substitute [tex]\(x = -1\)[/tex] into [tex]\( y = 3x - 8 \)[/tex]:
[tex]\[
y = 3(-1) - 8 = -3 - 8 = -11
\][/tex]
2. Substitute [tex]\(x = -1\)[/tex] into [tex]\( y = -x + 4 \)[/tex]:
[tex]\[
y = -(-1) + 4 = 1 + 4 = 5
\][/tex]
The first equation is not satisfied, so System 3 does not have the solution [tex]\((-1, 5)\)[/tex].
### System 4:
[tex]\[
\begin{cases}
y = 3x - 8 \\
y = -x - 4
\end{cases}
\][/tex]
1. Substitute [tex]\(x = -1\)[/tex] into [tex]\( y = 3x - 8 \)[/tex]:
[tex]\[
y = 3(-1) - 8 = -3 - 8 = -11
\][/tex]
2. Substitute [tex]\(x = -1\)[/tex] into [tex]\( y = -x - 4 \)[/tex]:
[tex]\[
y = -(-1) - 4 = 1 - 4 = -3
\][/tex]
Both equations are not satisfied, so System 4 does not have the solution [tex]\((-1, 5)\)[/tex].
### Conclusion:
The system of equations that has the solution [tex]\((-1, 5)\)[/tex] is:
[tex]\[
\begin{cases}
y = 3x + 8 \\
y = -x + 4
\end{cases}
\][/tex]
which is System 1.
