The sum of all values of [0, ] for which the system of linear equations x + y + z = 1 2x + 3y + 5z = 2 4x + 7y + (13 + 2 )z = 3 + 2 has no solution, is

Step-by-step solution

For the system to have no solution, we must have = 0 and at least one of _x, _y, _z 0 . = vmatrix 1 & 1 & 1 2 & 3 & 5 4 & 7 & 13 + 2 vmatrix Expanding along the first row: = 1(39 + 3 2 - 35) - 1(26 + 2 2 - 20) + 1(14 - 12) = (4 + 3 2 ) - (6 + 2 2 ) + 2 = 2 Setting = 0 gives 2 = 0 . For [0, ] , 2 [0, 2 ] , which gives 2 = 2 , 3 2 . Thus, = 4 or = 3 4 . Now we evaluate _z : _z = vmatrix 1 & 1 & 1 2 & 3 & 2 4 & 7 & 3 + 2 vmatrix _z = 1(9 + 3 2 - 14) - 1(6 + 2 2 - 8) + 1(14 - 12) _z = (-5 + 3 2 ) - (-2 + 2 2 ) + 2 = 2