To determine which system of equations has a solution of [tex]\( (1, 3, -1) \)[/tex], we need to substitute these values into each system and check if all equations in that system are satisfied.
System A:
[tex]\[
\begin{aligned}
-2x + 2y &= 4 \\
y + 7z &= -4 \\
x - y - z &= 4
\end{aligned}
\][/tex]
Substituting [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex], and [tex]\( z = -1 \)[/tex]:
[tex]\[
\begin{aligned}
-2(1) + 2(3) &= -2 + 6 = 4, \, \text{(True)} \\
3 + 7(-1) &= 3 - 7 = -4, \, \text{(True)} \\
1 - 3 - (-1) &= 1 - 3 + 1 = -1, \, \text{(False)}
\end{aligned}
\][/tex]
System A is not satisfied, as the third equation is false.
System B:
[tex]\[
\begin{aligned}
3x - y &= 8 \\
2y - z &= 5 \\
x - y + z &= -1
\end{aligned}
\][/tex]
Substituting [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex], and [tex]\( z = -1 \)[/tex]:
[tex]\[
\begin{aligned}
3(1) - 3 &= 3 - 3 = 0, \, \text{(False)} \\
2(3) - (-1) &= 6 + 1 = 7, \, \text{(False)} \\
1 - 3 + (-1) &= 1 - 3 - 1 = -3, \, \text{(False)}
\end{aligned}
\][/tex]
System B is not satisfied, as none of the equations are true.
System C:
[tex]\[
\begin{aligned}
x + 2y &= 7 \\
y + 2z &= 1 \\
x - y - z &= -1
\end{aligned}
\][/tex]
Substituting [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex], and [tex]\( z = -1 \)[/tex]:
[tex]\[
\begin{aligned}
1 + 2(3) &= 1 + 6 = 7, \, \text{(True)} \\
3 + 2(-1) &= 3 - 2 = 1, \, \text{(True)} \\
1 - 3 - (-1) &= 1 - 3 + 1 = -1, \, \text{(True)}
\end{aligned}
\][/tex]
System C is fully satisfied, as all three equations are true.
System D:
[tex]\[
\begin{aligned}
x + y &= 4 \\
y - z &= 2 \\
x + y - z &= -1
\end{aligned}
\][/tex]
Substituting [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex], and [tex]\( z = -1 \)[/tex]:
[tex]\[
\begin{aligned}
1 + 3 &= 1 + 3 = 4, \, \text{(True)} \\
3 - (-1) &= 3 + 1 = 4, \, \text{(False)} \\
1 + 3 - (-1) &= 1 + 3 + 1 = 5, \, \text{(False)}
\end{aligned}
\][/tex]
System D is not satisfied, as the second and third equations are false.
Hence, the system of equations that has a solution of [tex]\( (1, 3, -1) \)[/tex] is:
C.
[tex]\[
\begin{aligned}
x + 2y &= 7 \\
y + 2z &= 1 \\
x - y - z &= -1
\end{aligned}
\][/tex]
