what condition, if any, must a, b, and c
satisfy for the linear system to be consistent?
27. x + 3y − z = a
x + y + 2z = b
2y − 3z = c
For the given linear system to be consistent, it needs to satisfy the condition that the system has either a unique solution or infinitely many solutions. In terms of the coefficients a, b, and c, the system will be consistent if and only if the determinant of the coefficient matrix is not equal to zero.\n\nThe coefficient matrix A for the given system is:\n\n| 1  3  -1 |\n| 1  1   2 |\n| 0  2  -3 |\n\nThe system is consistent if and only if the determinant of A is not equal to zero. Therefore, the condition for the system to be consistent is that:\n\ndet(A) = 1(1*(-3) - 2*2) - 3(1*2 - 0*(-3)) + (-1)(1*2 - 0*1) ≠ 0\n\nSolving this determinant, we get:\n\ndet(A) = 1*(-3 - 4) - 3*(2) + (-1)*(2) = -7 - 6 - 2 ≠ 0\ndet(A) = -15 ≠ 0\n\nSo, the condition for the linear system to be consistent is that the determinant of the coefficient matrix A is not equal to zero (i.e., det(A) ≠ 0).