If the system of equations x+y+z=5, x+2y+3z=9, x+3y+αz=β has infinitely many solutions, then β−α equals?

D=|1 1 1|
|1 2 3|
|1 3 α| = |1 1 1|
|0 1 2|
|0 2 α-1| = (α-1-2) = (α-3)
for infinite solutions D=0 ⇒ α=3
Dx=0 ⇒ |5 1 1|
|9 2 3|
|β 3 α| = 0 ⇒ |0 1 1|
|4 2 3|
|β-5 3 α| = 0 ⇒ |0 1 1|
|4 1 2|
|β-5 0 α-3| = 0 ⇒ 2(β-5) - (α-3) = 0
β-5 - 2(α-3)/2 = 0
on α=3, we get
β-5 = 0
β = 5
we get Dy=Dz=0 α=3, β=13
α=3, β=13
β-α = 13-3 = 10