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X + 2 y = 3 a x+ by = 4

If b ≠ 2a, then there is exactly one solution.

Let a = k and b = 2 k => the system of equations becomes: x+2y = 3 and x+2y = 4/k

They will not have any solution if a = k and b = 2k , with k ≠ 4/3. That is if a=1, b=2, then there is no solution. For a=2, b=4 there is no solution.. for a=3, b=6 there is no solution.

If k = 4/3, or a= 4/3 and b=8/3, then there are infinite number of solutions.

reason: In that case of a=k, and b=2k, we get x+2y=3 and kx+2ky=4 or x+2y = 4/k => k must be 3. ============================== Another way, matrix [ 1 2 ] should be singular. [ a b ]