Example 1 - calculation not required
It can be observed that the values of a and b in the top equation are half of the bottom equation, but this is not true for the values of c. This implies that the ratios of a and b are same but different than that of c. Hence, the system has no solution.
Example 2 - mental math for those good with dividing fractions
It can be observed that the values of a, b, and c in the top equation are six times of the bottom equation. 3 is six times of 1/2, 2 is six times of 1/3, and 6 is six times of 1. This implies that the ratios of a, b, and c are same. Hence, the system has infinitely many solutions.