Question

Find the values of a and b for which the following system of linear equations is...

Find the values of a and b for which the following system of linear equations is (i) inconsistent; (ii) has a unique solution; (iii) has infinitely many solutions. For the case where the system has infinitely many solutions, write the general solution.

x + y + z = a

x + 2y ? z = 0

x + by + 3z = 2

Homework Answers

Answer #1

The augmented matrix of the given linear system is A (say) =

1

1

1

a

1

2

-1

0

1

b

3

2

To solve the given linear system, we will reduce A to its RREF as under:

1.Add -1 time row 1 to row 2

2.Add -1 time row 1 to row 3

3.Add -1 time row 2 to row 1

4.Add (1-b) times row 2 to row 3

5. Multiply row 3 by 1/2b

6.Add -3 time row 3 to row 1

7.Add 2 times row 3 to row 2

Then the RREF of A is

1

0

0

(ab+6a-6)/2b

0

1

0

2(1-a)/b

0

0

1

(a/2)-(a/b)+(1/b)

(i). In view of the 5th row operation and RREF of A, the given linear system will be inconsistent if b = 0. (ii).In view of the 5th row operation and the RREF of A, the given linear system has a unique solution if b is not equal to 0.

(ii). In view of the RREF of A, the given linear system cannot have infinite solutions.

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