The number of positive integral solutions of the equation x 3 + 1 x 2 y x 2 z x y 2 y 3 + 1 y 2 z x z 2 y z 2 z 3 + 1 = 11 is/are

Step-by-step solution

Multiplying R 1 by x , R 2 by y and R 3 by z and dividing the determinant by x y z , we get, 1 x y z x 4 + x x 3 y x 3 z x y 3 y 4 + y y 3 z x z 3 y z 3 z 4 + z = 11 ⇒ x y z x y z x 3 + 1 x 3 x 3 y 3 y 3 + 1 y 3 z 3 z 3 z 3 + 1 = 11 Use R 1 → R 1 + R 2 + R 3 x 3 + y 3 + z 3 + 1 1 1 1 y 3 y 3 + 1 y 3 z 3 z 3 z 3 + 1 = 11 ⇒ x 3 + y 3 + z 3 = 10 (as the value of the determinant is 1 ) Hence, all the possible solutions are ( 2 , 1 , 1 ) , ( 1 , 2 , 1 ) , ( 1 , 1 , 2 )