Investigate for what values of ๐ ๐๐๐
๐ the equation x+y+z=6; x+2y+3z=10; x+2y+๐z=๐ have

(i)no solution,

(ii) a unique solution,

(iii) infinite no. of solution.

#### Solution

Given eqn : x+y+z=6, x+2y+3z=10, x+2y+๐z=๐

A X = B

`[(1,1,1),(1,2,3),(1,2,lambda)][(x),(y),(z)]=[(6),(10),(mu)]`

Argumented matrix is :`[(1,1,1),(1,2,3),(1,2,lambda)][(6),(10),(mu)]`

`R_1-R_2,`

`->[(1,1,1,|,6),(0,1,2 ,|,4 ),(0,1,lambda-3,|,mu-6)]`

`R_2-R_1,`

`-> [(1,1,1,|,6),(0,1,2 ,|,4 ),(0,1,lambda-1,|,mu-10)]`

(i) When ๐=3, ๐≠๐๐ ๐๐๐๐ ๐(๐)=๐,๐(๐จโฎ๐ฉ)=๐

r(A)≠๐(๐จโฎ๐ฉ)

Hence for ๐=3 , ๐≠๐๐ system is inconsistent.

No solution exist.

(ii) When ๐≠3,๐≠๐๐ ,๐(๐จ)=๐(๐จโฎ๐ฉ)=๐

Unique solution exist.

(iii) When ๐=3,๐=๐๐ ๐(๐จ)=๐(๐จโฎ๐ฉ)=๐<๐

Infinite solution.