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Sagot :
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]
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]
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