Let S be the set of all values of [0, 2 ] for which the system of linear equations x + y + z = 1 x + y + z = 1 x - y + z 2 = 0 has no solution. Then the value of 4 _ S is equal to

Step-by-step solution

For the system to have no solution, the determinant of the coefficient matrix D must be zero, and at least one of D_x, D_y, D_z must be non-zero. D = vmatrix & & 1 & & 1 1 & -1 & 2 vmatrix Expanding the determinant: D = ( 2 + 1) - ( 2 - 1) + 1 (- - ) D = ^2 2 + - ^2 2 + - - D = 2 ( ^2 - ^2 ) = - 2 2 = - 1 2 4 Setting D = 0 4 = 0 4 = n = n 4 . In [0, 2 ] , the possible values of are 0, 4 , 2 , 3 4 , , 5 4 , 3 2 , 7 4 , 2 . Now we evaluate D_x : D_x = vmatrix 1 & & 1 1 & & 1 0 & -1 & 2 vmatrix = 1( 2 + 1) - ( 2 ) + 1