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?
First equation: 3a + 4b = 7
Simplifying the second equation: 6a + 4b = k - 3a - 8b
9a + 12b = k
To have infinitely many solutions, get the coefficients of a and b to be the same:
first equation: 3a + 4b = 7 ---> x3 ---> 9a + 12b = 21
second equation: 9a + 12 b = k
There will be infinitely many solutions when k = 21.
(There will be none if k isn't 21.)