For a certain value of k, the system

3a + 4b = 7,

6a + 4b = k - 3a - 8b

has infinitely many solutions (a,b). What is k?

Our two equations are:

3a + 4b = 7

9a + 12b = k

If (a,b) have infinite many solutions, then those two equations above should be redundant.

If they are redundant, then the second equation is equal to the first equation by a factor of x3.

Thus, k = 7 x 3 = 21.