If the system of linear equations : aligned & x+y+2 z=6 & 2 x+3 y+ a z= a +1 & -x-3 y+ b z=2 ~b aligned where a, b R , has infinitely many solutions, then 7 a+3 b is equal to :

Step-by-step solution

The given equations are aligned & x+y+2 z=6 & 2 x+3 y+a z=a+1 & -x-3 y+b z=2 b, where a, b, R aligned For infinite many solutions: D=D₁=D₂=D₃=0 aligned & D= | array ccc 1 & 1 & 2 2 & 3 & a -1 & -3 & b array |=2 a+b-6 & D₁= | array ccc 6 & 1 & 2 a+1 & 3 & a 2 b & -3 & b array |=12 a+5 b+a b-6 & D₂= | array ccc 1 & 6 & 2 2 & a+1 & a -1 & 2 b & b array |=-4 a-3 b-a b+2 & and D₃= | array ccc 1 & 1 & 6 2 & 3 & a+1 -1 & -3 & 2 b array |=2 a+2 b-16 aligned from above relations aligned & a=-2, b=10 & 7 a+3 b=16 aligned