Given the system of equations in two variables a_1 x+b_1 y=c_1 and a_2 x+b_2 y=c_2...

 

assign possible values of a_1, b_1, c_1, a_2, b_2, c_2 to make the system inconsistent and independent

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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` .

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