# x1+2x2-3x3=a x1+x2-2x3=b 2x1-x2-3x3=c under what condition on a,b and c does the system have a solution?( use rref)

x1+2x2-3x3=a
x1+x2-2x3=b
2x1-x2-3x3=c
under what condition on a,b and c does the system have a solution?( use rref)
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The system will have a solution irrespective of the values of a,b and c(assuming that a,b and c are constants).
The solution would be
x1 = (-15a + 21b + c)/8
x2 = (a - 3b + c)/8
x3 = (-7a + 5b + c)/8
In case a,b and c are not constants and are dependant on x1 or x2 or x3, then the system will have a solution only when the coefficient matrix (obtained after rearranging the terms so that only constants are on RHS) is invertible. It can be easily checked by just checking the determinant of the coefficient matrix. If the determinant of the matrix turns out to be 0, then the matrix in not invertible and hence the system of equations will not have a solution. however if the determinant of the coefficient matrix is non zero then the inverse of the matrix exists and hence the system of equations will have a solution.