If the system of simultaneous linear equations (x+y+z=a, x-y+b z=2 ), (2 x+3 y-z=1 ) has infinitely many solutions, then (b-5 a= )

Step-by-step solution

Given that system of linear equations ( aligned x+y+z & =a (i) x-y+b z & =2 (ii) 2 x+3 y-z & =1 (iii) aligned ) has infinitely many solutions, ( aligned & [ array ccc 1 & 1 & 1 1 & -1 & b 2 & 3 & -1 array ]=0 & 1(1-3 b)-1(-1-2 b)+1(3+2=0 & 1-3 b+1+2 b+5=0 & & -b=-7 & & b=7 aligned ) Adding Eqs. (i) and (ii), we get ( aligned 2 x+8 z & =a+2 x+4 z & = a+2 2 (iv) aligned ) Multiply Eq. (i) by 3 then subtract Eq. (iii) from it, ( aligned 3 x+3 y+3 z & =3 a -2 x 3 y z & =-1 x+4 z & =3 a-1 (v) aligned ) By Eqs. (iv) and