Are the following pair of linear equations consistent? Justify your answer.

(i) - 3x - 4y = 12 and 4y + 3x = 12

(ii)

(iii) 2ax + by = a and 4ax + 2by - 2a = 0; a, b ≠ 0

(iv) x + 3y = 11 and 2x + 6y = 11

Conditions for pair of linear equations are consistent

a₁/a₂ b₁/b_2. [unique solution]

and a₁/a₂ = b₁/b₂ = c₁/c₂ [coincident or infinitely many solutions]

(i) No.

The given pair of linear equations -

- 3x - 4y - 12 = 0 and 4y + 3x - 12 = 0

Comparing with ax + by + c = 0;

Here, a₁ = - 3, b₁ = - 4, c₁ = - 12;

And a₂ = 3, b₂ = 4, c₂ = - 12;

a₁ /a₂ = - 3/3 = - 1

b₁ /b₂ = - 4/4 = - 1

c₁ /c₂ = - 12/ - 12 = 1

Here, a₁/a₂ = b₁/b₂ c₁/c₂

Hence, the pair of linear equations has no solution, i.e., inconsistent.

(ii) Yes.

The given pair of linear equations

and

Comparing with ax + by + c = 0;

Here, a₁ = 3/5, b₁ = - 1, c₁ = - 1/2;

And a₂ = 1/5, b₂ = 3, c₂ = - 1/6;

a₁ /a₂ = 3

b₁ /b₂ = - 1/ - 3 = 1/3

c₁ /c₂ = 3

Here, a₁/a₂ b₁/b_2.

Hence, the given pair of linear equations has unique solution, i.e., consistent.

(iii) Yes.

The given pair of linear equations -

2ax + by –a = 0 and 4ax + 2by - 2a = 0

Comparing with ax + by + c = 0;

Here, a₁ = 2a, b₁ = b, c₁ = - a;

And a₂ = 4a, b₂ = 2b, c₂ = - 2a;

a₁ /a₂ = 1/2

b₁ /b₂ = 1/2

c₁ /c₂ = 1/2

Here, a₁/a₂ = b₁/b₂ = c₁/c₂

Hence, the given pair of linear equations has infinitely many solutions, i.e., consistent or dependent.

(iv) No.

The given pair of linear equations

x + 3y = 11 and 2x + 6y = 11

Comparing with ax + by + c = 0;

Here, a₁ = 1, b₁ = 3, c₁ = 11

And a₂ = 2, b₂ = 6, c₂ = 11

a₁ /a₂ = 1/2

b₁ /b₂ = 1/2

c₁ /c₂ = 1

Here, a₁/a₂ = b₁/b₂ c₁/c₂.

Hence, the given pair of linear equations has no solution.