I.
Examine whether the following systems of equations are consistent or inconsistent and if consistent find the complete solutions,
Question 1.
x + y + z = 4
2x + 5y – 2z = 3
x + 7y – 7z = 5

Answer:
Augmented matrix of the above system is

Rank of the matrix ρ(A) = 2 and ρ(AB) = 3.
Since ρ(A) ≠ ρ(AB), the given system of equa¬tions are inconsistent.

Question 2.
x + y + z = 6
x – y + z = 2
2x – y + 3z = 9

Answer:
Augmented matrix [AB] = $\left[\begin{array}{rrrr} 1 & 1 & 1 & 6 \\ 1 & -1 & 1 & 2 \\ 2 & -1 & 3 & 9 \end{array}\right]$
Apply operations R₂ – R₁, R₃ – 2R₁, we get

Here ρ(A) = 3 and ρ(AB) = 3
Since ρ(A) = ρ(AB), the given system is consistent and has a unique solution.

Question 3.
x + y + z = 1
2x + y + z = 2
x + 2y + 2z = 1 (March 2015-T.S)

Answer:
Augmented matrix of the system is

ρ(AB) = 2 and ρ(A) = 2 and ρ(A) = ρ(AB) < 3
The given system of equations is consis¬tent and has infinitely many solutions.
The given system is equivalent to x + y + z = 1 and y + z = 0.
Solution set is

Question 4.
x + y + z = 9
2x + 5y + 7z = 52
2x + y – z = 0

Answer:
Augmented matrix of the system

Here ρ(A) = ρ(AB) = 3; and the system of given equations is consistent; and has a unique solution.
Also x = 1, y = 3, z = 5 form the solution.

Question 5.
x + y + z = 6
x + 2y + 3z = 10
x + 2y + 4z = 1

Answer:
Augmented matrix of the system is

Question 6.
x – 3y – 8z = – 10
3x + y – 4z = 0
2x + 5y + 6z = 13

Answer:
The augmented matrix of the above system is

Since ρ(A) = 2 = ρ(AB) < 3, given system of equations is consistent with infinitely many solutions.
The given system is equivalent to
x – 3y – 8z = – 10,
y + 2z = 3
Put z = t then y = 3 – 2t
∴ x = – 10 + 3(3 – 2t) + 8t
= -10 + 9 – 6t + 8t
= 2t – 1
Hence the solutions are given by
x = 2t – 1, y = 3 – 2t and z = t
Where t is any scalar.

Question 7.
2x + 3y + z = 9
x + 2y + 3z = 6
3x + y + 2z = 8

Answer:
Augmented matrix of the above system is

ρ(A) = ρ(AB) = 3; system is consistent and has a unique solution given by
x = $\frac{35}{18}$, y = $\frac{29}{18}$, z = $\frac{5}{18}$

Question 8.
x + y + 4z = 6
3x + 2y – 2z = 9
5x + y + 2z = 13

Answer:
Augmented matrix of the system

ρ(A) = ρ(AB) = 3;
Hence the system is consistent and has a unique solution given by
x = 2, y = 2, z = $\frac{1}{2}$.