You need to impose the following conditions for the system to be inconsistent and independent determinant of matrix formed from coefficients of variables such that:

`Delta = [[a_1,b_1],[a_2,b_2]] = 0`

`Delta = [[a_1,b_1],[a_2,b_2]] = a_1*b_2 - b_1*a_2`

`a_1*b_2 - b_1*a_2 = 0 =gt a_1*b_2 =b_1*a_2 =gt a_1/a_2 = b_1/b_2`

You also need to impose the condition for caracteristic determinant of system such that:

`C = [[a_1,c_1],[a_2,c_2]] != 0`

`a_1/a_2 != c_1/c_2`

**Hence, evaluating the relations between coefficients `a_1,a_2,b_1,b_2,c_1,c_2` for the system to be inconsistent and independent yields `a_1/a_2 = b_1/b_2 != c_1/c_2` .**