On comparing the ratios a1/a2, b1/b2 and c1/c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:

(i) 5x – 4y + 8 = 0
7x + 6y – 9 = 0

(ii) 9x + 3y + 12 = 0
18x + 6y + 24 = 0

(iii) 6x – 3y + 10 = 0
2x – y + 9 = 0

Solution:

(i) Given expressions;

5x − 4y + 8 = 0

7x + 6y − 9 = 0

Comparing these equations with a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 we get,

a₁ = 5, b₁ = -4, c₁ = 8

a₂ = 7, b₂ = 6, c₂ = -9

(a₁/a₂) = 5/7

(b₁/b₂) = -4/6 = -2/3

(c₁/c₂) = 8/-9

Since, (a₁/a₂) ≠ (b₁/b₂)

So, the pairs of equations given in the question have a unique solution and the lines cross each other at exactly one point.

(ii) Given expressions;

9x + 3y + 12 = 0

18x + 6y + 24 = 0

Comparing these equations with a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 we get,

a₁ = 9, b₁ = 3, c₁ = 12

a₂ = 18, b₂ = 6, c₂ = 24

(a₁/a₂) = 9/18 = 1/2

(b₁/b₂) = 3/6 = 1/2

(c₁/c₂) = 12/24 = 1/2

Since (a₁/a₂) = (b₁/b₂) = (c₁/c₂)

So, the pairs of equations given in the question have infinite possible solutions and the lines are coincident.

(iii) Given Expressions;

6x – 3y + 10 = 0

2x – y + 9 = 0

Comparing these equations with a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 we get,

a₁ = 6, b₁ = -3, c₁ = 10

a₂ = 2, b₂ = -1, c₂ = 9

(a₁/a₂) = 6/2 = 3/1

(b₁/b₂) = -3/-1 = 3/1

(c₁/c₂) = 10/9

Since (a₁/a₂) = (b₁/b₂) ≠ (c₁/c₂)

So, the pairs of equations given in the question are parallel to each other and the lines never intersect each other at any point and there is no possible solution for the given pair of equations.