Comparing the ratio and find out whether the following pairs of linear equations are consistent or inconsistent.

(i) 2x – 3y = 8; 4x – 6y = 9

(ii) 3x – y = 2; 6x – 2y = 4

(iii) 2x – 2y = 2; 4x – 4y = 5

(iv)

For any pair of linear equations to be consistent or inconsistent, we

check the following situations:

a.

In this case, the pair of linear equations is consistent. This means there is unique solution for the given pair of linear equations. The graph of linear equations would be two intersecting lines.

b.

In this case, the pair of linear equations is inconsistent. This means there is no solution for the given pair of linear equations. The graph of linear equations will be two parallel lines.

c.

In this case, the pair of linear equations is dependent and consistent. This means there are infinitely many solutions for the given pair of linear equations. The graph of linear equations will be coincident lines.

(i) 2x – 3y = 8; 4x – 6y = 9

Here, a₁ = 2, b₁ = – 3, c₁ = 8

and a₂ = 4, b₂ = – 6, c₂ = 9

Clearly,

Hence, the given lines are parallel. So the given pair of equation has no solution and it is inconsistent.

(ii) 3x – y = 2; 6x – 2y = 4

Here, a₁ = 3, b₁ = – 1, c₁ = 2

and a₂ = 6, b₂ = – 2, c₂ = 4

Clearly,

Hence, the given lines are dependent and consistent. So the given pair of equation has infinitely many solutions.

(iii) 2x – 2y = 2; 4x – 4y = 5

Here, a₁ = 2, b₁ = – 2, c₁ = 2

and a₂ = 4, b₂ = – 4, c₂ = 5

Clearly,

Hence, the given lines are parallel. So the given pair of equation has no solution and it is inconsistent.

(iv)

Here, , b₁ = 2, c₁ = 8

and a₂ = 2, b₂ = 3, c₂ = 12

Clearly,

Hence, the given lines are two intersecting lines. So the given pair of equation has unique solution and it is consistent.