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P Investigate for What Values of ๐ "๐’‚๐’๐’…" ๐€ the Equations : 2 X + 3 Y + 5 Z = 9 7 X + 3 Y โˆ’ 2 Z = 8 2 X + 3 Y + ฮป Z = ฮผ Have (I) No Solution (Ii) Unique Solution (Iii) Infinite Value - Applied Mathematics 1

Investigate for what values of ๐ "๐’‚๐’๐’…" ๐€ the equations : `2x+3y+5z=9`

`7x+3y-2z=8`

`2x+3y+λz=μ`

Have (i) no solution (ii) unique solution (iii) Infinite value 

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Solution

Given eqn : 

`2x+3y+5z=9` 

`7x+3y-2z=8`

`2x+3y+λz=μ` 

    `  A x=B`

∴ `[[2,3,5],[7,3,-2],[2,3,λ]] [[x],[y],[z]] =[[9],[8],[μ]]` 

Argumented matrix is : `[[2,3,5],[7,3,-2],[2,3,λ]][[9],[8],[μ]]`

`R_3-R_1`  

→` [[2,3,5],[7,3,-2],[0,0,λ-5]]  [[6],[4],[μ-9]]`

(i) When ๐œ†=5,๐≠๐Ÿ— ๐’•๐’‰๐’†๐’ ๐’“(๐’‚)= )=๐Ÿ,๐’“(๐‘จโ‹ฎ๐‘ฉ)=๐Ÿ‘
r(A)≠๐’“(๐‘จโ‹ฎ๐‘ฉ)
No Solution.
(ii) When ๐œ†≠5,๐≠๐Ÿ— ,๐’“(๐‘จ)=๐’“(๐‘จโ‹ฎ๐‘ฉ)=๐Ÿ‘
Unique solution exist.
(iii) When ๐œ†=5,๐=๐Ÿ— ๐’“(๐‘จ)=๐’“(๐‘จโ‹ฎ๐‘ฉ)=๐Ÿ<๐Ÿ‘
Infinite solution.

Concept: Reduction to Normal Form
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