If the planes x + y + z = 1 , 2x + 3y + 4z = 5 , and 4x + 7y + z = intersect in a single line, then the value of + is

Step-by-step solution

For the three planes to intersect in a single line, the system of linear equations must have infinitely many solutions. This requires = 0 and _x = _y = _z = 0 . Evaluating : = vmatrix 1 & 1 & 1 2 & 3 & 4 4 & 7 & vmatrix = 1(3 - 28) - 1(2 - 16) + 1(14 - 12) = - 10 Setting = 0 gives = 10 . Now, evaluating _z : _z = vmatrix 1 & 1 & 1 2 & 3 & 5 4 & 7 & vmatrix = 1(3 - 35) - 1(2 - 20) + 1(14 - 12) _z = - 13 Setting _z = 0 gives = 13 . Therefore, + = 10 + 13 = 23 . Answer: 23