# P Investigate for What Values of ๐ "๐๐๐" ๐ the Equations : 2 X + 3 Y + 5 Z = 9 7 X + 3 Y โ 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

#### 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 ๐๐๐๐ ๐(๐)= )=๐,๐(๐จโฎ๐ฉ)=๐
r(A)≠๐(๐จโฎ๐ฉ)
No Solution.
(ii) When ๐≠5,๐≠๐ ,๐(๐จ)=๐(๐จโฎ๐ฉ)=๐
Unique solution exist.
(iii) When ๐=5,๐=๐ ๐(๐จ)=๐(๐จโฎ๐ฉ)=๐<๐
Infinite solution.

Concept: Reduction to Normal Form
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