Linear algebra / pivot position (conceptual)

Solve the non-homogeneous system
x(1) + 2x(2) = 3
2x(1) + 4x(2) = 6

After you reduce the matrix represented by this system, you get
[1 2 | 3]
[0 0 | 0]

My question is, why is there a solution, namely
[3] + x(2)[-2]
[0]______[1]

if there is no pivot in the reduced augmented matrix given above? I thought if there isn't a pivot in every row, then b is not a solution to a matrix equation?

Edit: Does a solution exist to the above reduced matrix given above because the theorem-for which every row in a matrix must have a pivot in its row otherwise b isn't a solution-only applies for some b, not just all b's? And therefore, the reduced matrix above is a legitimate solution because the matrix doesn't violate the existence and uniqueness theorem?