# Investigate for what values of 𝝁 𝒂𝒏𝒅 𝝀 the equation x+y+z=6; x+2y+3z=10; x+2y+𝜆z=𝝁 have (i)no solution, (ii) a unique solution, (iii) infinite no. of solution. - Applied Mathematics 1

Sum

Investigate for what values of 𝝁 𝒂𝒏𝒅 𝝀 the equation x+y+z=6; x+2y+3z=10; x+2y+𝜆z=𝝁 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+𝜆z=𝝁

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 𝜆=3, 𝝁≠𝟏𝟎 𝒕𝒉𝒆𝒏 𝒓(𝒂)=𝟐,𝒓(𝑨⋮𝑩)=𝟑
r(A)≠𝒓(𝑨⋮𝑩)
Hence for 𝜆=3 , 𝝁≠𝟏𝟎 system is inconsistent.
No solution exist.
(ii) When 𝜆≠3,𝝁≠𝟏𝟎 ,𝒓(𝑨)=𝒓(𝑨⋮𝑩)=𝟑
Unique solution exist.
(iii) When 𝜆=3,𝝁=𝟏𝟎 𝒓(𝑨)=𝒓(𝑨⋮𝑩)=𝟐<𝟑
Infinite solution.

Concept: consistency and solutions of homogeneous and non – homogeneous equations
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