Answer :
Let's match each system of equations to its solution represented by an augmented matrix.
### Systems of Equations and Solutions:
1. System 1:
[tex]\[ \begin{array}{c} x + y + z = 1,100 \\ x - 2y - z = -500 \\ 2x + 3y + 2z = 2,600 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 750 \\ 0 & 1 & 0 & 850 \\ 0 & 0 & 1 & 1,000 \end{array}\right] \][/tex]
2. System 2:
[tex]\[ \begin{array}{c} x + y + z = 2,600 \\ x + y - z = 600 \\ 2x + y + 2z = 3,050 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 1,300 \\ 0 & 1 & 0 & 1,100 \\ 0 & 0 & 1 & 900 \end{array}\right] \][/tex]
3. System 3:
[tex]\[ \begin{array}{c} x + y + z = 1,900 \\ x - y - 2z = -2,000 \\ 2x + 2y + z = 1,100 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 500 \\ 0 & 1 & 0 & 400 \\ 0 & 0 & 1 & 200 \end{array}\right] \][/tex]
4. System 4:
[tex]\[ \begin{array}{c} x + y + z = 1,900 \\ x - y - 2z = -2,000 \\ 2x + 2y + z = 2,900 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 400 \\ 0 & 1 & 0 & 600 \\ 0 & 0 & 1 & 900 \end{array}\right] \][/tex]
5. System 5:
[tex]\[ \begin{array}{c} x + y + z = 3,300 \\ x + y - z = 1,500 \\ 2x + 3y + 2z = 7,700 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 1300 \\ 0 & 1 & 0 & 1100 \\ 0 & 0 & 1 & 900 \end{array}\right] \][/tex]
### Matching:
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 750 \\ 0 & 1 & 0 & 850 \\ 0 & 0 & 1 & 1,000\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=1,100 \\ x-2 y-z=-500 \\ 2 x+3 y+2z=2,600 \end{array}\)[/tex]
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 1,300 \\ 0 & 1 & 0 & 1,100 \\ 0 & 0 & 1 & 900\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=2,600 \\ x+y-z=600 \\ 2 x+y+2 z=3,050 \end{array}\)[/tex]
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 500 \\ 0 & 1 & 0 & 400 \\ 0 & 0 & 1 & 200\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=1,900 \\ x-y-2 z=-2,000 \\ 2 x+2y+z=1,100 \end{array}\)[/tex]
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 400 \\ 0 & 1 & 0 & 600 \\ 0 & 0 & 1 & 900\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=1,900 \\ x-y-2 z=-2,000 \\ 2x+2y+z=2,900 \end{array}\)[/tex]
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 1,300 \\ 0 & 1 & 0 & 1,100 \\ 0 & 0 & 1 & 900\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=3,300 \\ x+y-z=1,500 \\ 2 x+3 y+2 z=7,700 \end{array}\)[/tex]
These are the correct matchings between the systems of equations and their solutions.
### Systems of Equations and Solutions:
1. System 1:
[tex]\[ \begin{array}{c} x + y + z = 1,100 \\ x - 2y - z = -500 \\ 2x + 3y + 2z = 2,600 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 750 \\ 0 & 1 & 0 & 850 \\ 0 & 0 & 1 & 1,000 \end{array}\right] \][/tex]
2. System 2:
[tex]\[ \begin{array}{c} x + y + z = 2,600 \\ x + y - z = 600 \\ 2x + y + 2z = 3,050 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 1,300 \\ 0 & 1 & 0 & 1,100 \\ 0 & 0 & 1 & 900 \end{array}\right] \][/tex]
3. System 3:
[tex]\[ \begin{array}{c} x + y + z = 1,900 \\ x - y - 2z = -2,000 \\ 2x + 2y + z = 1,100 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 500 \\ 0 & 1 & 0 & 400 \\ 0 & 0 & 1 & 200 \end{array}\right] \][/tex]
4. System 4:
[tex]\[ \begin{array}{c} x + y + z = 1,900 \\ x - y - 2z = -2,000 \\ 2x + 2y + z = 2,900 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 400 \\ 0 & 1 & 0 & 600 \\ 0 & 0 & 1 & 900 \end{array}\right] \][/tex]
5. System 5:
[tex]\[ \begin{array}{c} x + y + z = 3,300 \\ x + y - z = 1,500 \\ 2x + 3y + 2z = 7,700 \end{array} \][/tex]
Solution:
[tex]\[ \left[\begin{array}{ccc|c} 1 & 0 & 0 & 1300 \\ 0 & 1 & 0 & 1100 \\ 0 & 0 & 1 & 900 \end{array}\right] \][/tex]
### Matching:
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 750 \\ 0 & 1 & 0 & 850 \\ 0 & 0 & 1 & 1,000\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=1,100 \\ x-2 y-z=-500 \\ 2 x+3 y+2z=2,600 \end{array}\)[/tex]
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 1,300 \\ 0 & 1 & 0 & 1,100 \\ 0 & 0 & 1 & 900\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=2,600 \\ x+y-z=600 \\ 2 x+y+2 z=3,050 \end{array}\)[/tex]
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 500 \\ 0 & 1 & 0 & 400 \\ 0 & 0 & 1 & 200\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=1,900 \\ x-y-2 z=-2,000 \\ 2 x+2y+z=1,100 \end{array}\)[/tex]
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 400 \\ 0 & 1 & 0 & 600 \\ 0 & 0 & 1 & 900\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=1,900 \\ x-y-2 z=-2,000 \\ 2x+2y+z=2,900 \end{array}\)[/tex]
- [tex]\(\left[\begin{array}{ccc|c}1 & 0 & 0 & 1,300 \\ 0 & 1 & 0 & 1,100 \\ 0 & 0 & 1 & 900\end{array}\right]\)[/tex] matches with:
* [tex]\(\begin{array}{c} x+y+z=3,300 \\ x+y-z=1,500 \\ 2 x+3 y+2 z=7,700 \end{array}\)[/tex]
These are the correct matchings between the systems of equations and their solutions.
