Let's analyze the systems step by step to answer both questions.
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To get System \( B \) from System \( A \):
System \( A \) is: \[ \begin{cases} -4x - 6y = 9 \ 3x + y = -4 \end{cases} \]
System \( B \) is: \[ \begin{cases} -4x - 6y = 9 \ -x - 5y = 5 \end{cases} \]
The first equation in both systems, \(-4x - 6y = 9\), is identical. The second equations differ:
- From System \( A \): \( 3x + y = -4 \)
- From System \( B \): \( -x - 5y = 5 \)
Now, let's see if we can express the second equation of System \( B \) in terms of the two equations from System \( A \):
If we take the sum of the left-hand sides of both equations in System \( A \): \[ (-4x - 6y) + (3x + y) = 9 + (-4) \] This simplifies to: \[ -4x - 6y + 3x + y = 5 \ -x - 5y = 5 \] We can indeed replace the second equation in System \( A \) with the sum of both equations to get the second equation in System \( B\).
Therefore, the correct answer is: (Choice B) Replace one equation with the sum/difference of both equations.
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Are the two systems equivalent — do they have the same solution?
Since we established that we can transform System \( A \) into System \( B \) using a valid algebraic operation, both systems effectively contain the same information about the variables \( x \) and \( y \). Therefore, they share the same solution.
The answer is: (Choice A) Yes.
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