Given : - 

Two equation 3x + y = 5 and – 6x – 2y = 9 

Tip : - We know that 

For a system of 2 simultaneous linear equation with 2 unknowns 

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

 x = \(\frac{D_1}{D}\), y = \(\frac{D_2}{D}\)

(ii) If D = 0 and D₁ = D₂ = 0, then the system is consistent and has infinitely many solution. 

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

Now, 

We have, 

3x + y = 5

– 6x – 2y = 9 

Lets find D

⇒ D = \(\begin{vmatrix} 3& 1 \\[0.3em] -6 &-2 \\[0.3em] \end{vmatrix}\) 

⇒ D = – 6 – 6 

⇒ D = 0

Again, 

D₁ by replacing 1^st column by B 

Here,

B = \(\begin{vmatrix} 5 \\[0.3em] 9\\[0.3em] \end{vmatrix}\)

⇒ D₁ = \(\begin{vmatrix} 5& 1 \\[0.3em] 9 &-2 \\[0.3em] \end{vmatrix}\) 

⇒ D₁ = – 10 – 9 

⇒ D₁ = – 19 

And, 

D₂ by replacing 2^nd column by B 

Here,

B = \(\begin{vmatrix} 5 \\[0.3em] 9\\[0.3em] \end{vmatrix}\)

⇒ D₂ = \(\begin{vmatrix} 3& 5 \\[0.3em] -6 &9 \\[0.3em] \end{vmatrix}\) 

⇒ D₂ = 27 + 30 

⇒ D₂ = 57 

So, here we can see that 

D = 0 and D₁ and D₂ are non – zero 

Hence the given system of equation is inconsistent.