Consider this system:

2x + (m-1)y = 3

(m+1)x + 4y = -3

If the determinant is not zero then by Cramer's Rule there exist a unique solution, thus the determinant has to be zero:

(m^2-1)-8=0 thus m^2-9=0 thus m=-3 or 3.

When, m=3 we have:

2x+2y=3

4x+4y=-3

Which is equivalent to:

4x+4y=6

4x+4y=-3

An impossibility thus at m=3 there are no solutions.

If m=-3

2x - 4y = 3

-2x + 4y = -3

Which is equivalent to:

2x - 4y = 3

2x - 4y = 3

Which has infinitely many solutions.

Thus at m=-3 there are infinitely mant solutions.