Show that each of the following systems of linear equations is inconsistent:

x + y + z = 3

2x – y + z = 2

3x + 6y + 5z = 20.

Given: - Three equation

x + y + z = 3

2x – y + z = 2

3x + 6y + 5z = 20.

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

3x + 6y + 5z = 20.

Lets find D

⇒

Expanding along 1^st row

⇒ D = 1[ – 5 – 1(6)] – (1)[(5)2 – 3] + 1[12 + 3]

⇒ D = 1[ – 11] – 1[7] + 1[15]

⇒ D = – 3

So, here we can see that

D ≠ 0

Hence the given system of equation is consistent.