Final answer:
The question contains a misunderstanding in its phrasing as it suggests a transformation T taking a vector ⟨x, y⟩ to ⟨-2, 7⟩, then showing it should be equal to ⟨x, y⟩, which is contradictory. This cannot be solved with the presented options, and additional context or correction of the question is required.
Step-by-step explanation:
The student is asking about the condition under which a transformation T maps a vector ⟨x, y⟩ to another vector ⟨-2, 7⟩. According to the transformation provided, when we apply T to ⟨x, y⟩, we get the vector ⟨-2, 7⟩. This corresponds to two equations: the first element after transformation being equal to -2 and the second element being equal to 7. Thus, for T⟨-2,7⟩(x,y) = (3,-1), the implication is that when the transformation is applied to vector ⟨-2, 7⟩, it yields ⟨x, y⟩ as ⟨-2, 7⟩. However, we have a contradiction here because according to the transformation rule, applying T to ⟨x, y⟩ should yield ⟨-2, 7⟩, not ⟨-2, 7⟩ turning into ⟨x, y⟩.
It appears there is a misunderstanding in the way the question is posed. It should probably be that application of T to ⟨x, y⟩ yields ⟨-2, 7⟩, suggesting that x = -2 and y = 7, which is not one of the provided answer options. Or the question should present T as a transformation that takes ⟨x, y⟩ to ⟨-2, 7⟩, which might be a linear transformation involving addition or multiplication factors. Nevertheless, based on the current question's phrasing, there seems to be a typo, and it cannot be solved as stated.