# Investigate for what values of ๐ ๐๐๐ ๐ the equation x+y+z=6; x+2y+3z=10; x+2y+๐z=๐ have (i)no solution, (ii) a unique solution, (iii) infinite no. of solution. - Applied Mathematics 1

Sum

Investigate for what values of ๐ ๐๐๐ ๐ the equation x+y+z=6; x+2y+3z=10; x+2y+๐z=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+๐z=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≠๐๐ ๐๐๐๐ ๐(๐)=๐,๐(๐จโฎ๐ฉ)=๐
r(A)≠๐(๐จโฎ๐ฉ)
Hence for ๐=3 , ๐≠๐๐ system is inconsistent.
No solution exist.
(ii) When ๐≠3,๐≠๐๐ ,๐(๐จ)=๐(๐จโฎ๐ฉ)=๐
Unique solution exist.
(iii) When ๐=3,๐=๐๐ ๐(๐จ)=๐(๐จโฎ๐ฉ)=๐<๐
Infinite solution.

Concept: consistency and solutions of homogeneous and non โ homogeneous equations
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