The value of `lamda` for which the system of equations :
`x+y+z=6,x+2y=3z=10,x+4y+lamdaz=12` has a unique solution is :

Explanation

we can write the above system of equations in the matrix form
`[(1,1,1),(1,2,3),(1,2,lambda)][(,x),(,y),(,z)]=[(,6),(,10),(,mu)]`
`rArr AX=B`
where A`A=[(1,1,1),(1,2,3),(1,2,lambda)],X=[(,x),(,y),(,z)]andB=[(,6),(,10),(,mu)]`
`therefore` augmented matrix
`C=[A:B]=[(1,1,1,vdots,6),(1,2,3,vdots,10),(1,2,lambda,vdots,mu)]`
Appying `R_(2) to R_(2) and R_(3) to R_(3)-R_(1),` we get
`C=[(1,1,1,vdots,6),(0,1,2,vdots,4),(0,1,lambda-1,vdots,mu-6)]`
Applying `R_(3) to R_(3)-R_(2),` we get
(I) No solution `p(A)!=p(c)`
`i.e.," " lambda-3!=0 and mu-10!= 0`
(ii) A unique solution `p(A) = p(c) =3`
`ie.," " lambda-3and mu in R`
`therefore" " lambda!=3 and mu in R`
(iii) Infinite number of solutions `p(A)=p(c)(angle3)`
`ie.," " lambda-3=0 and mu -10=0`
`therefore " " lambda=3 and mu =10`