Question

Test the consistency of the following equations. Find the solutions if consistent:
(i) 3x + 3y + 2z = 1, x + 2y = 4, 10y + 3z = -2, 2x – 3y – z = 5.

(ii) 2x + y + z + u = 2, 3x - y + z – u = 2, x + 2y – z + u = 1, 6x + 2y + z + u = 5.

pls answer with full explanation


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Answer 1

Answer:

After making two zeros and expanding, we get

△=

∣

∣

∣

∣

∣

∣

∣

∣

1

1

3

1

3

λ+2

1

−2

−3

∣

∣

∣

∣

∣

∣

∣

∣

=3(λ−5)

△

x

=

∣

∣

∣

∣

∣

∣

∣

∣

1

λ

2λ+1

1

3

2λ+2

1

−2

−3

∣

∣

∣

∣

∣

∣

∣

∣

=(λ−5)(λ+2)

△

y

=

∣

∣

∣

∣

∣

∣

∣

∣

1

1

3

1

λ

2λ+1

1

−2

−3

∣

∣

∣

∣

∣

∣

∣

∣

=0

△

z

=

∣

∣

∣

∣

∣

∣

∣

∣

1

1

3

1

3

λ+2

1

λ

2λ+1

∣

∣

∣

∣

∣

∣

∣

∣

=−(λ−1)(λ−5)

△=0 i.e, λ=5 the system has unique solution given by

△x

x

=

△

y

y

=

△

z

z

=

△

1

Hence the system has infinite solutions.

Putting λ=5 and eliminating z, we have 3x+5y=7 i.e only one equation in two variables.

Putting x=c,y=

5

7−3c

and hence from any z=−

5

2(1+c)