Answer :
To determine which systems of equations have no solution, we need to analyze each system one-by-one. Let's verify:
1. System 1:
[tex]\[ \begin{aligned} x + 4y & = 23 \\ -3x - 12y & = -1 \end{aligned} \][/tex]
After examination, this system has no solution.
2. System 2:
[tex]\[ \begin{aligned} 2x + 4y & = 22 \\ -x - 2y & = 11 \end{aligned} \][/tex]
After examination, this system has no solution.
3. System 3:
[tex]\[ \begin{aligned} 2x + y & = 17 \\ -4x - 2y & = -34 \end{aligned} \][/tex]
After examination, this system has a solution.
4. System 4:
[tex]\[ \begin{aligned} 3y & = 10 - x \\ 2x + 6y & = 7 \end{aligned} \][/tex]
After examination, this system has no solution.
5. System 5:
[tex]\[ \begin{aligned} 2x + y & = 15 \\ x & = 15 - 2y \end{aligned} \][/tex]
After examination, this system has a solution.
6. System 6:
[tex]\[ \begin{aligned} y & = 13 - 2x \\ 4x - y & = -1 \end{aligned} \][/tex]
After examination, this system has a solution.
Based on this detailed analysis, the systems of equations that have no solution are:
1. System 1:
[tex]\[ \begin{aligned} x + 4y & = 23 \\ -3x - 12y & = -1 \end{aligned} \][/tex]
2. System 2:
[tex]\[ \begin{aligned} 2x + 4y & = 22 \\ -x - 2y & = 11 \end{aligned} \][/tex]
4. System 4:
[tex]\[ \begin{aligned} 3y & = 10 - x \\ 2x + 6y & = 7 \end{aligned} \][/tex]
1. System 1:
[tex]\[ \begin{aligned} x + 4y & = 23 \\ -3x - 12y & = -1 \end{aligned} \][/tex]
After examination, this system has no solution.
2. System 2:
[tex]\[ \begin{aligned} 2x + 4y & = 22 \\ -x - 2y & = 11 \end{aligned} \][/tex]
After examination, this system has no solution.
3. System 3:
[tex]\[ \begin{aligned} 2x + y & = 17 \\ -4x - 2y & = -34 \end{aligned} \][/tex]
After examination, this system has a solution.
4. System 4:
[tex]\[ \begin{aligned} 3y & = 10 - x \\ 2x + 6y & = 7 \end{aligned} \][/tex]
After examination, this system has no solution.
5. System 5:
[tex]\[ \begin{aligned} 2x + y & = 15 \\ x & = 15 - 2y \end{aligned} \][/tex]
After examination, this system has a solution.
6. System 6:
[tex]\[ \begin{aligned} y & = 13 - 2x \\ 4x - y & = -1 \end{aligned} \][/tex]
After examination, this system has a solution.
Based on this detailed analysis, the systems of equations that have no solution are:
1. System 1:
[tex]\[ \begin{aligned} x + 4y & = 23 \\ -3x - 12y & = -1 \end{aligned} \][/tex]
2. System 2:
[tex]\[ \begin{aligned} 2x + 4y & = 22 \\ -x - 2y & = 11 \end{aligned} \][/tex]
4. System 4:
[tex]\[ \begin{aligned} 3y & = 10 - x \\ 2x + 6y & = 7 \end{aligned} \][/tex]