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Investigate for what values of ๐ ๐’‚๐’๐’… ๐€ the equation x+y+z=6; x+2y+3z=10; x+2y+๐œ†z=๐ have (i)no solution, (ii) a unique solution, (iii) infinite no. of solution. - Applied Mathematics 1

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Sum

Investigate for what values of ๐ ๐’‚๐’๐’… ๐€ the equation x+y+z=6; x+2y+3z=10; x+2y+๐œ†z=๐ have
(i)no solution,
(ii) a unique solution,
(iii) infinite no. of solution.

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Solution

Given eqn : x+y+z=6, x+2y+3z=10,  x+2y+๐œ†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œ†=3, ๐≠๐Ÿ๐ŸŽ ๐’•๐’‰๐’†๐’ ๐’“(๐’‚)=๐Ÿ,๐’“(๐‘จโ‹ฎ๐‘ฉ)=๐Ÿ‘
r(A)≠๐’“(๐‘จโ‹ฎ๐‘ฉ)
Hence for ๐œ†=3 , ๐≠๐Ÿ๐ŸŽ system is inconsistent.
No solution exist.
(ii) When ๐œ†≠3,๐≠๐Ÿ๐ŸŽ ,๐’“(๐‘จ)=๐’“(๐‘จโ‹ฎ๐‘ฉ)=๐Ÿ‘
Unique solution exist.
(iii) When ๐œ†=3,๐=๐Ÿ๐ŸŽ ๐’“(๐‘จ)=๐’“(๐‘จโ‹ฎ๐‘ฉ)=๐Ÿ<๐Ÿ‘
Infinite solution.

Concept: consistency and solutions of homogeneous and non โ€“ homogeneous equations
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