An ordered pair α ,   β for which the system of linear equations 1 + α x + β y + z = 2 α x + 1 + β y + z = 3 α x + β y + 2 z = 2 has a unique solution, is :

Step-by-step solution

For unique solution 1 + α β 1 α 1 + β 1 α β 2 ≠ 0 Applying C 1 → C 1 + C 2 + C 3 , we get α + β + 2 β 1 α + β + 2 1 + β 1 α + β + 2 β 2 ≠ 0 ⇒ α + β + 2 1 β 1 1 1 + β 1 1 β 2 ≠ 0 ⇒ α + β + 2 ≠ 0 Clearly, point 2 ,   4 satisfying the given condition.