If the determinant of the coefficient matrix is not zero, then the system IS CONSISTENT.


Hence, the only case, when it can be inconsistent, is equality of the determinant to zero.


It gives  2*3 - (-5)*k = 0,  or  6 + 5k = 0,   5k = -6,   k =  = -1.2.


With this value of k, the system takes the form


     2x   - 5y = -1.2      (1)

    -1.2x + 3y =  1        (2)



To convince yourself that it is INCONSISTENT, multiply equation (1) by 3; multiply equation (2) by -5.

You will get then the EQUIVALENT system


    6x - 15y = -3.6     (3)

    6x - 15y =  -5.     (4)



Comparing equations (3) and (4), you see that their left sides are IDENTICAL, 

while their right sides are DIFFERENT.


It shows that the system (3), (4) is INCOMPATIBLE  (= inconsistent).


With it, the equivalent system (1), (2) is inconsistent, too.