Investigate for what values of ๐ "๐๐๐ " 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=5,๐≠๐ ๐๐๐๐ ๐(๐)= )=๐,๐(๐จโฎ๐ฉ)=๐

r(A)≠๐(๐จโฎ๐ฉ)

No Solution.

(ii) When ๐≠5,๐≠๐ ,๐(๐จ)=๐(๐จโฎ๐ฉ)=๐

Unique solution exist.

(iii) When ๐=5,๐=๐ ๐(๐จ)=๐(๐จโฎ๐ฉ)=๐<๐

Infinite solution.

Concept: Reduction to Normal Form

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