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To match each system of equations to the inverse of its coefficient matrix, [tex]\( A^{-1} \)[/tex], and the matrix of its solution, [tex]\( X \)[/tex], we will use the given information to establish the correct pairs.
### System of Equations 1:
[tex]\[ \begin{array}{c} x + y + z = 1600 \\ x - 2y - z = -1000 \\ 2x + 3y + 2z = 3600 \end{array} \][/tex]
### System of Equations 2:
[tex]\[ \begin{array}{c} x + y + z = 2600 \\ x + y - z = 600 \\ 2x + y + 2z = 4350 \end{array} \][/tex]
### Possible Solutions:
#### Solution 1:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} -1.5 & 0.5 & 1 \\ 2 & 0 & -1 \\ 0.5 & -0.5 & 0 \end{array}\right], \\ X = \left[\begin{array}{c} -550 \\ 2150 \\ 1000 \end{array}\right] \end{array} \][/tex]
#### Solution 2:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} 1.5 & 0.5 & -0.5 \\ -2.5 & -0.5 & 1.5 \\ 2 & 0 & -1 \end{array}\right], \\ X = \left[\begin{array}{c} 1300 \\ -2100 \\ 2700 \end{array}\right] \end{array} \][/tex]
### Matching the Systems:
- First system of equations matches with:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} -1.5 & 0.5 & 1 \\ 2 & 0 & -1 \\ 0.5 & -0.5 & 0 \end{array}\right], \\ X = \left[\begin{array}{c} -550 \\ 2150 \\ 1000 \end{array}\right] \end{array} \][/tex]
- Second system of equations matches with:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} 1.5 & 0.5 & -0.5 \\ -2.5 & -0.5 & 1.5 \\ 2 & 0 & -1 \end{array}\right], \\ X = \left[\begin{array}{c} 1300 \\ -2100 \\ 2700 \end{array}\right] \end{array} \][/tex]
### Result:
1. For the system of equations:
[tex]\[ \begin{array}{c} x + y + z = 1600 \\ x - 2y - z = -1000 \\ 2x + 3y + 2z = 3600 \end{array} \][/tex]
the correct pair is:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} -1.5 & 0.5 & 1 \\ 2 & 0 & -1 \\ 0.5 & -0.5 & 0 \end{array}\right] \\ X = \left[\begin{array}{c} -550 \\ 2150 \\ 1000 \end{array}\right] \end{array} \][/tex]
2. For the system of equations:
[tex]\[ \begin{array}{c} x + y + z = 2600 \\ x + y - z = 600 \\ 2x + y + 2z = 4350 \end{array} \][/tex]
the correct pair is:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} 1.5 & 0.5 & -0.5 \\ -2.5 & -0.5 & 1.5 \\ 2 & 0 & -1 \end{array}\right] \\ X = \left[\begin{array}{c} 1300 \\ -2100 \\ 2700 \end{array}\right] \end{array} \][/tex]
### System of Equations 1:
[tex]\[ \begin{array}{c} x + y + z = 1600 \\ x - 2y - z = -1000 \\ 2x + 3y + 2z = 3600 \end{array} \][/tex]
### System of Equations 2:
[tex]\[ \begin{array}{c} x + y + z = 2600 \\ x + y - z = 600 \\ 2x + y + 2z = 4350 \end{array} \][/tex]
### Possible Solutions:
#### Solution 1:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} -1.5 & 0.5 & 1 \\ 2 & 0 & -1 \\ 0.5 & -0.5 & 0 \end{array}\right], \\ X = \left[\begin{array}{c} -550 \\ 2150 \\ 1000 \end{array}\right] \end{array} \][/tex]
#### Solution 2:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} 1.5 & 0.5 & -0.5 \\ -2.5 & -0.5 & 1.5 \\ 2 & 0 & -1 \end{array}\right], \\ X = \left[\begin{array}{c} 1300 \\ -2100 \\ 2700 \end{array}\right] \end{array} \][/tex]
### Matching the Systems:
- First system of equations matches with:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} -1.5 & 0.5 & 1 \\ 2 & 0 & -1 \\ 0.5 & -0.5 & 0 \end{array}\right], \\ X = \left[\begin{array}{c} -550 \\ 2150 \\ 1000 \end{array}\right] \end{array} \][/tex]
- Second system of equations matches with:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} 1.5 & 0.5 & -0.5 \\ -2.5 & -0.5 & 1.5 \\ 2 & 0 & -1 \end{array}\right], \\ X = \left[\begin{array}{c} 1300 \\ -2100 \\ 2700 \end{array}\right] \end{array} \][/tex]
### Result:
1. For the system of equations:
[tex]\[ \begin{array}{c} x + y + z = 1600 \\ x - 2y - z = -1000 \\ 2x + 3y + 2z = 3600 \end{array} \][/tex]
the correct pair is:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} -1.5 & 0.5 & 1 \\ 2 & 0 & -1 \\ 0.5 & -0.5 & 0 \end{array}\right] \\ X = \left[\begin{array}{c} -550 \\ 2150 \\ 1000 \end{array}\right] \end{array} \][/tex]
2. For the system of equations:
[tex]\[ \begin{array}{c} x + y + z = 2600 \\ x + y - z = 600 \\ 2x + y + 2z = 4350 \end{array} \][/tex]
the correct pair is:
[tex]\[ \begin{array}{c} A^{-1} = \left[\begin{array}{ccc} 1.5 & 0.5 & -0.5 \\ -2.5 & -0.5 & 1.5 \\ 2 & 0 & -1 \end{array}\right] \\ X = \left[\begin{array}{c} 1300 \\ -2100 \\ 2700 \end{array}\right] \end{array} \][/tex]
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