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Select ALL the correct answers.

Select all the statements that are true for the following systems of equations.

[tex]\[
\begin{array}{ccc}
\text{System A} & \text{System B} & \text{System C} \\
2x - 3y = 4 & 3x - 4y = 5 & 2x - 3y = 4 \\
4x - y = 18 & y = 5x + 3 & 12x - 3y = 54 \\
\end{array}
\][/tex]

System C:
[tex]\[
\begin{array}{c}
2x - 3y = 4 \\
12x - 3y = 54 \\
\end{array}
\][/tex]

- Systems B and C have the same solution.
- System C simplifies to [tex]\( 2x - 3y = 4 \)[/tex] and [tex]\( 4x - y = 18 \)[/tex] by dividing the second equation by three.
- Systems A and B have different solutions.
- All three systems have different solutions.
- Systems A and C have the same solution.

Sagot :

GinnyAnswer
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.
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