Answer :
Of course! Let's match each system of equations with its respective solution:
1. System of Equations:
[tex]\[ \begin{aligned} 2x + y & = 12 \\ x & = 9 - 2y \end{aligned} \][/tex]
Solution: [tex]\(\{x: 5, y: 2\}\)[/tex]
Explanation: This system's solution is [tex]\(x = 5\)[/tex] and [tex]\(y = 2\)[/tex].
___
2. System of Equations:
[tex]\[ \begin{aligned} y & = 11 - 2x \\ 4x - 3y & = -13 \end{aligned} \][/tex]
Solution: [tex]\(\{x: 2, y: 7\}\)[/tex]
Explanation: This system's solution is [tex]\(x = 2\)[/tex] and [tex]\(y = 7\)[/tex].
___
3. System of Equations:
[tex]\[ \begin{aligned} y & = 10 + x \\ -3x + 3y & = 30 \end{aligned} \][/tex]
Solution: [tex]\(\{x = y - 10\}\)[/tex]
Explanation: This system has an infinite set of solutions given by the relationship [tex]\(x = y - 10\)[/tex].
___
4. System of Equations:
[tex]\[ \begin{aligned} x + 3y & = 16 \\ 2x - y & = 11 \end{aligned} \][/tex]
Solution: [tex]\(\{x: 7, y: 3\}\)[/tex]
Explanation: This system's solution is [tex]\(x = 7\)[/tex] and [tex]\(y = 3\)[/tex].
___
5. System of Equations:
[tex]\[ \begin{aligned} 2x + y & = 11 \\ x - 2y & = -7 \end{aligned} \][/tex]
Solution: [tex]\(\{x: 3, y: 5\}\)[/tex]
Explanation: This system's solution is [tex]\(x = 3\)[/tex] and [tex]\(y = 5\)[/tex].
___
For completeness, one system of equations does not have a solution listed here and is not part of the results. It is:
System (No Solution Listed):
[tex]\[ \begin{aligned} x + 2y & = 9 \\ 2x + 4y & = 20 \end{aligned} \][/tex]
To summarize the matching pairs:
1.
[tex]\[ \begin{aligned} 2x + y & = 12 \\ x & = 9 - 2y \end{aligned} \][/tex]
matches with [tex]\(\{x : 5, y : 2\}\)[/tex]
2.
[tex]\[ \begin{aligned} y & = 11 - 2x \\ 4x - 3y & = -13 \end{aligned} \][/tex]
matches with [tex]\(\{x : 2, y : 7\}\)[/tex]
3.
[tex]\[ \begin{aligned} y & = 10 + x \\ -3x + 3y & = 30 \end{aligned} \][/tex]
matches with [tex]\(\{x : y - 10\}\)[/tex]
4.
[tex]\[ \begin{aligned} x + 3y & = 16 \\ 2x - y & = 11 \end{aligned} \][/tex]
matches with [tex]\(\{x : 7, y : 3\}\)[/tex]
5.
[tex]\[ \begin{aligned} 2x + y & = 11 \\ x - 2y & = -7 \end{aligned} \][/tex]
matches with [tex]\(\{x : 3, y : 5\}\)[/tex]
1. System of Equations:
[tex]\[ \begin{aligned} 2x + y & = 12 \\ x & = 9 - 2y \end{aligned} \][/tex]
Solution: [tex]\(\{x: 5, y: 2\}\)[/tex]
Explanation: This system's solution is [tex]\(x = 5\)[/tex] and [tex]\(y = 2\)[/tex].
___
2. System of Equations:
[tex]\[ \begin{aligned} y & = 11 - 2x \\ 4x - 3y & = -13 \end{aligned} \][/tex]
Solution: [tex]\(\{x: 2, y: 7\}\)[/tex]
Explanation: This system's solution is [tex]\(x = 2\)[/tex] and [tex]\(y = 7\)[/tex].
___
3. System of Equations:
[tex]\[ \begin{aligned} y & = 10 + x \\ -3x + 3y & = 30 \end{aligned} \][/tex]
Solution: [tex]\(\{x = y - 10\}\)[/tex]
Explanation: This system has an infinite set of solutions given by the relationship [tex]\(x = y - 10\)[/tex].
___
4. System of Equations:
[tex]\[ \begin{aligned} x + 3y & = 16 \\ 2x - y & = 11 \end{aligned} \][/tex]
Solution: [tex]\(\{x: 7, y: 3\}\)[/tex]
Explanation: This system's solution is [tex]\(x = 7\)[/tex] and [tex]\(y = 3\)[/tex].
___
5. System of Equations:
[tex]\[ \begin{aligned} 2x + y & = 11 \\ x - 2y & = -7 \end{aligned} \][/tex]
Solution: [tex]\(\{x: 3, y: 5\}\)[/tex]
Explanation: This system's solution is [tex]\(x = 3\)[/tex] and [tex]\(y = 5\)[/tex].
___
For completeness, one system of equations does not have a solution listed here and is not part of the results. It is:
System (No Solution Listed):
[tex]\[ \begin{aligned} x + 2y & = 9 \\ 2x + 4y & = 20 \end{aligned} \][/tex]
To summarize the matching pairs:
1.
[tex]\[ \begin{aligned} 2x + y & = 12 \\ x & = 9 - 2y \end{aligned} \][/tex]
matches with [tex]\(\{x : 5, y : 2\}\)[/tex]
2.
[tex]\[ \begin{aligned} y & = 11 - 2x \\ 4x - 3y & = -13 \end{aligned} \][/tex]
matches with [tex]\(\{x : 2, y : 7\}\)[/tex]
3.
[tex]\[ \begin{aligned} y & = 10 + x \\ -3x + 3y & = 30 \end{aligned} \][/tex]
matches with [tex]\(\{x : y - 10\}\)[/tex]
4.
[tex]\[ \begin{aligned} x + 3y & = 16 \\ 2x - y & = 11 \end{aligned} \][/tex]
matches with [tex]\(\{x : 7, y : 3\}\)[/tex]
5.
[tex]\[ \begin{aligned} 2x + y & = 11 \\ x - 2y & = -7 \end{aligned} \][/tex]
matches with [tex]\(\{x : 3, y : 5\}\)[/tex]
