If the system of equations  ax+y+z=0,  x+by+z=0,  x+y+cz=0  (a,b,c≠1)has a non-zero solution. Then11−a+11−b+11−c=

The condition for a homogeneous system of equations to have a non-zero solution is Δ=0⇒a111b111c=0      R3→R3-R2, R2→R2-R1 ⇒a111-ab-1001-bc-1=0 ⇒ab-1c-1-11-ac-1+11-a1-b=0 ⇒a1-b1-c+11-a1-c+11-a1-b=0 ⇒a1-a+11-b+11-c=0   dividing through out by 1-a1-b1-c ⇒1+a-11-a+11-b+11-c=0⇒11−a+11−b+11−c=1