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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?

 May 3, 2023
 #1
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This can only have infinitely many solutions if both equations say the same thing. Let's work with the second one a bit.

6a+2b=k+3a+mb

3a+2b=k+mb

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

k+mb=2.

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.

 May 3, 2023

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