Final answer:
There is an error in the question as the given pair of equations cannot have infinitely many solutions since they do not represent the same line, and no value of k is provided in the question that would result in the equations being identical.
Step-by-step explanation:
The value of k for which the pair of equations x=y−2 and 6=2y−3 has infinitely many solutions is determined by ensuring both equations represent the same line. To do this, we can rewrite the second equation in terms of x. Starting from 6=2y−3, we add 3 to both sides and then divide by 2 to isolate y, obtaining y = (6 + 3) / 2.
Simplifying, we get y = 4.5. However, this cannot be equal to x due to the discrepancy introduced by the first equation, x=y−2. Therefore, there can only be infinitely many solutions if the two equations are identical, which they are not, regardless of the value of k.
Thus, there seems to be an error in the question since none of the provided options for k would result in the two equations having infinitely many solutions.
First, we set x = y - 2 and 6 = 2y - 3 equal to each other:
y - 2 = 2y - 3
4 = y
Now we substitute this value of y back into either of the original equations:
x = 4 - 2 = 2
Therefore, the value of k that gives infinitely many solutions is (a) k = 3.