If the system of equations x + y + z = 5 , x + 2 y + 3 z = 9 , x + 3 y + α z = β has inifinitely many solutions, then β - α equals

Step-by-step solution

The system of linear equations a 1 x + b 1 y + c 1 z + d 1 = 0 ,   a 2 x + b 2 y + c 2 z + d 2 = 0 and a 3 x + b 3 y + c 3 z + d 3 = 0 have infinitely many solutions, if D = a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 = 0 ,   D 1 = d 1 b 1 c 1 d 2 b 2 c 2 d 3 b 3 c 3 = 0 ,   D 2 = a 1 d 1 c 1 a 2 d 2 c 2 a 3 d 3 c 3 = 0 and D 3 = a 1 b 1 d 1 a 2 b 2 d 2 a 3 b 3 d 3 = 0 . Hence, for the system of equations x + y + z = 5 ,   x + 2 y + 3 z = 9 and x + 3 y + α z = β have infinitely many solution,