Answer :
Given the systems of equations, we will determine the solutions for Systems A, B, and C, and then check which statements are true.
### System A:
[tex]\[ \begin{cases} 2x - 3y = 4 \\ 4x - y = 18 \end{cases} \][/tex]
By solving System A, we get the solution:
[tex]\[ x = 5, \; y = 2 \][/tex]
### System B:
[tex]\[ \begin{cases} 3x - 4y = 5 \\ y - 5x = 3 \end{cases} \][/tex]
By solving System B, we get the solution:
[tex]\[ x = -1, \; y = -2 \][/tex]
### System C:
[tex]\[ \begin{cases} 2x - 3y = 4 \\ 12x - 3y = 54 \end{cases} \][/tex]
By solving System C, we get the solution:
[tex]\[ x = 5, \; y = 2 \][/tex]
Now, let's analyze the given statements based on the solutions we found:
1. Systems B and C have the same solution.
- System B has the solution [tex]\( x = -1, \; y = -2 \)[/tex].
- System C has the solution [tex]\( x = 5, \; y = 2 \)[/tex].
Since the solutions are not the same, this statement is false.
2. System C simplifies to [tex]\(2x - 3y = 4\)[/tex] and [tex]\(4x - y = 18\)[/tex] by dividing the second equation by three.
Though not required for validation, let’s check:
- Original System C:
[tex]\[ \begin{cases} 2x - 3y = 4 \\ 12x - 3y = 54 \end{cases} \][/tex]
- Dividing the second equation by 3:
[tex]\[ \begin{cases} 2x - 3y = 4 \\ 4x - y = 18 \end{cases} \][/tex]
This is indeed System A. Thus, this statement would be true.
3. Systems A and B have different solutions.
- System A has the solution [tex]\( x = 5, \; y = 2 \)[/tex].
- System B has the solution [tex]\( x = -1, \; y = -2 \)[/tex].
Since the solutions are different, this statement is true.
4. All three systems have different solutions.
- System A and System C both have the solution [tex]\( x = 5, \; y = 2 \)[/tex].
- System B has the solution [tex]\( x = -1, \; y = -2 \)[/tex].
Since Systems A and C have the same solution, this statement is false.
5. Systems A and C have the same solution.
- System A has the solution [tex]\( x = 5, \; y = 2 \)[/tex].
- System C has the solution [tex]\( x = 5, \; y = 2 \)[/tex].
Since the solutions are the same, this statement is true.
Therefore, the correct statements are:
- Systems A and B have different solutions.
- Systems A and C have the same solution.
### System A:
[tex]\[ \begin{cases} 2x - 3y = 4 \\ 4x - y = 18 \end{cases} \][/tex]
By solving System A, we get the solution:
[tex]\[ x = 5, \; y = 2 \][/tex]
### System B:
[tex]\[ \begin{cases} 3x - 4y = 5 \\ y - 5x = 3 \end{cases} \][/tex]
By solving System B, we get the solution:
[tex]\[ x = -1, \; y = -2 \][/tex]
### System C:
[tex]\[ \begin{cases} 2x - 3y = 4 \\ 12x - 3y = 54 \end{cases} \][/tex]
By solving System C, we get the solution:
[tex]\[ x = 5, \; y = 2 \][/tex]
Now, let's analyze the given statements based on the solutions we found:
1. Systems B and C have the same solution.
- System B has the solution [tex]\( x = -1, \; y = -2 \)[/tex].
- System C has the solution [tex]\( x = 5, \; y = 2 \)[/tex].
Since the solutions are not the same, this statement is false.
2. System C simplifies to [tex]\(2x - 3y = 4\)[/tex] and [tex]\(4x - y = 18\)[/tex] by dividing the second equation by three.
Though not required for validation, let’s check:
- Original System C:
[tex]\[ \begin{cases} 2x - 3y = 4 \\ 12x - 3y = 54 \end{cases} \][/tex]
- Dividing the second equation by 3:
[tex]\[ \begin{cases} 2x - 3y = 4 \\ 4x - y = 18 \end{cases} \][/tex]
This is indeed System A. Thus, this statement would be true.
3. Systems A and B have different solutions.
- System A has the solution [tex]\( x = 5, \; y = 2 \)[/tex].
- System B has the solution [tex]\( x = -1, \; y = -2 \)[/tex].
Since the solutions are different, this statement is true.
4. All three systems have different solutions.
- System A and System C both have the solution [tex]\( x = 5, \; y = 2 \)[/tex].
- System B has the solution [tex]\( x = -1, \; y = -2 \)[/tex].
Since Systems A and C have the same solution, this statement is false.
5. Systems A and C have the same solution.
- System A has the solution [tex]\( x = 5, \; y = 2 \)[/tex].
- System C has the solution [tex]\( x = 5, \; y = 2 \)[/tex].
Since the solutions are the same, this statement is true.
Therefore, the correct statements are:
- Systems A and B have different solutions.
- Systems A and C have the same solution.
