Math, asked by aryangupta03042012, 3 months ago

Q. 12. Investigate for what values of a and B the simultaneous equations x + y + z = 6; x + 2y + 3z = 10;
x + 2y + az = B have No Solution, Unique Solution and Infinitely many solutions.​

Answers

Answered by khadijamussarat698
2

Answer:

(i) no solution (ii) a unique solution and (iii) an infinite number of solutions.

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ANSWER

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.

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