Question

Investigate for what values of and , the system of linear equations

+ + = 6, + 2 + 3 = 10, + 2 + =

have (i) no solution, (ii) unique solution and (iii) infinite numbe of solutions.​

Answer 1

Answer:

no solution is your answer

Step-by-step explanation:

The given system can be written as

⎣

⎢

⎢

⎡

1

1

1

1

2

2

1

3

λ

⎦

⎥

⎥

⎤

⎣

⎢

⎢

⎡

x

y

z

⎦

⎥

⎥

⎤

=

⎣

⎢

⎢

⎡

6

10

μ

⎦

⎥

⎥

⎤

AX=B

The augmented matrix.

(A,B)=

⎣

⎢

⎢

⎡

1

1

1

1

2

2

1

3

λ

6

10

μ

⎦

⎥

⎥

⎤

=

⎣

⎢

⎢

⎡

1

1

1

1

2

2

1

3

λ−3

6

4

μ−10

⎦

⎥

⎥

⎤

R

2

→R

2

−R

1

R

3

→R

3

−R

2

Case (i) : λ−3=0 and μ−10



=0

(ie) λ=3,μ



=10

ρ(A)=2,ρ(A,B)=3

∴ρ(A)



=ρ(A,B)

The given system is inconsistent but has no solution.

Case (ii) : λ−3



=0 and μ∈R

(ie) λ



=3

\rho(A) = \rho(A,B) =3$$

The given system is consistent and has unique solution.

Case (iii) : λ=3 and μ=10

ρ(A)=ρ(A,B)=2< number of unknowns.

The given system is consistent but has an infinite number of solutions.