If the system of linear equations x + 2y - 3z = 4 2x + 5y - 4z = 5 4x + 9y + z = has no solution, then which of the following can be the ordered pair ( , ) ?

Step-by-step solution

For the system to have no solution, we must have = 0 and at least one of ₁, ₂, ₃ 0 . = vmatrix 1 & 2 & -3 2 & 5 & -4 4 & 9 & vmatrix = 1(5 + 36) - 2(2 + 16) - 3(18 - 20) = + 10 . Setting = 0 gives = -10 . Now, evaluate ₃ : ₃ = vmatrix 1 & 2 & 4 2 & 5 & 5 4 & 9 & vmatrix = 1(5 - 45) - 2(2 - 20) + 4(18 - 20) = - 13 . For the system to be inconsistent (no solution), we require ₃ 0 , which implies - 13 0 13 . Hence, the conditions are = -10 and 13 . Among the given options, the pair (-10, 15) satisfies these conditions