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?

The first equation is: 3a + 4b = 7

The second equation is: 6a + 4b = k - 3a - 8b

which becomes: 9a + 12b = k

Multiplying the first equation by 3: 9a + 12b = 21

For these to have infinitely many solutions, these must be two forms for the same equation; so k = 21