# 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 consistent and independent

You need to remember that a consistent independent system has only one solution.

You should remember what the condition for a system to be consistent is: the determinant of coefficient matrix is not zero such that:

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

`Delta = a_1*b_2 - a_2*b_1`

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

`a_1/a_2 != b_1/b_2`

You may solve the system using Cramer's rule such that:

`x =([[c_1,b_1],[c_2,b_2]])/Delta ; y = ([[a_1,c_1],[a_2,c_2]])/Delta`

Notice that in determinant `[[c_1,b_1],[c_2,b_2]], ` the column of constant terms replaces the columns of coefficients of x and in determinant  `[[a_1,c_1],[a_2,c_2]], ` the column of constant terms replaces the columns of coefficients of y.

Hence, the system is consistent if you consider the relation between the coefficients of x and y variables `a_1/a_2 != b_1/b_2.`

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