Show that each of the following systems of linear equations has infinite number of solutions and solve:

x – y + z = 3

2x + y – z = 2

– x – 2y + 2z = 1

Given: - Three equation

x – y + z = 3

2x + y – z = 2

– x – 2y + 2z = 1

Tip: - We know that

For a system of 3 simultaneous linear equation with 3 unknowns

(i) If D ≠ 0, then the given system of equations is consistent and has a unique solution given by

(ii) If D = 0 and D₁ = D₂ = D₃ = 0, then the given system of equation may or may not be consistent. However if consistent, then it has infinitely many solution.

(iii) If D = 0 and at least one of the determinants D₁, D₂ and D₃ is non – zero, then the system is inconsistent.

Now,

We have,

x – y + z = 3

2x + y – z = 2

– x – 2y + 2z = 1

Lets find D

⇒

Expanding along 1^st row

⇒ D = 1[2 – ( – 1)( – 2)] – ( – 1)[(2)2 – (1)] + 1[ – 4 – ( – 1)]

⇒ D = 1[0] + 1[3] + [ – 3]

⇒ D = 0

Again, D₁ by replacing 1^st column by B

Here

⇒

⇒ D₁ = 3[2 – ( – 1)( – 2)] – ( – 1)[(2)2 – ( – 1)] + 1[ – 4 – 1]

⇒ D_{1 =} 3[2 – 2] + [4 + 1] + 1[ – 5]

⇒ D_{1 =} 0 + 5 – 5

⇒ D₁ = 0

Also, D₂ by replacing 2^nd column by B

Here

⇒

⇒ D₂ = 1[4 – ( – 1)(1)] – (3)[(2)2 – (1)] + 1[2 – ( – 2)]

⇒ D_{2 =} 1[4 + 1] – 3[4 – 1] + 1[4]

⇒ D_{2 =} 5 – 9 + 4

⇒ D₂ = 0

Again, D₃ by replacing 3^rd column by B

Here

⇒

⇒ D₃ = 1[1 – ( – 2)(2)] – ( – 1)[(2)1 – 2( – 1)] + 3[2( – 2) – 1( – 1)]

⇒ D_{3 =} [1 + 4] + [2 + 2] + 3[ – 4 + 1]

⇒ D_{3 =} 5 + 4 – 9

⇒ D₃ = 0

So, here we can see that

D = D₁ = D₂ = D₃ = 0

Thus,

Either the system is consistent with infinitely many solutions or it is inconsistent.

Now, by 1^st two equations, written as

x – y = 3 – z

2x + y = 2 + z

Now by applying Cramer’s rule to solve them,

New D and D₁, D₂

⇒

⇒ D = 1 + 2

⇒ D = 3

Again, D₁ by replacing 1^st column with

⇒

⇒ D₁ = 3 – z – ( – 1)(2 + z)

⇒ D₁ = 5

Again, D₂ by replacing 2^nd column with

⇒

⇒ D₂ = 2 + z – 2 (3 – z)

⇒ D₂ = – 4 + 3z

Hence, using Cramer’s rule

⇒

⇒

again,

⇒

⇒

Let, z = k

Then

And z = k

By changing value of k you may get infinite solutions