For certain values of k and m, the system
3a + 2b = 2
6a + 2b = k + 3a + mb
has infinitely many solutions (a,b). What are k and m?
This can only have infinitely many solutions if both equations say the same thing. Let's work with the second one a bit.
We now have each equation saying that 3a+2b is equal to something. If the two separate expressions are equal, then there will be infinitely many solutions. Set
If we make m=1, we get k+b=2, which varies based on b, so that cannot be it.
If we make m=0, we get k=2, which is a solution.