+24 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?
+1610 For the system to have infinitely many solutions, the left-hand sides must be equal to each other, regardless of the values of a and b. Thus, 3a + 2b = 6a + 2b, which implies a = 0. Substituting this value into the first equation yields 2b = 2. Hence, b = 1. Then, the second equation becomes 0 - mb = k - 3(0) - m(1), which simplifies to k = m.
Therefore, k = m = \boxed{1}.
+6 [1] 3a + 2b = 2
6a + 2b = k - 3a - ab
[2] 9a + (2 + m) * b = k
Equation [1] is equivalent to
[3] 9a + 6b = 6
There are an infinite number of solutions if [2] and [3] are equivalent. That is true only if the coefficients of b in both equations are the same and the constants in both are the same.
2+m = 6 --> m = 4
k = 6
ANSWERS: k = 6; m = 4
