Let the three planes P₁ : x + y + z = 1 , P₂ : x + 2y + z = 2 , and P₃ : x + 4y + ^2 z = intersect in a line. The sum of all possible values of ( + ) is equal to

Step-by-step solution

For the three planes to intersect in a line, the corresponding system of linear equations must have infinitely many solutions. This requires the determinant of the coefficient matrix, D , to be zero: D = vmatrix 1 & 1 & 1 1 & 2 & 1 & 4 & ^2 vmatrix = 0 Applying C₂ C₂ - C₁ and C₃ C₃ - C₁ : D = vmatrix 1 & 0 & 0 1 & 1 & - 1 1 & 3 & ^2 - 1 vmatrix = ( ^2 - 1) - 3( - 1) = ( - 1)( - 2) Thus, D = 0 = 1 or = 2 . For infinitely many solutions, we also need D_x = D_y = D_z = 0 . Case 1: = 1 D_x = vmatrix 1 & 1 & 1 2 & 2 & 1