For any pair of linear equation
a₁ x + b₁ y + c₁ = 0
a₂ x + b₂ y + c₂ = 0
a) a₁/a₂ ≠ b₁/b₂ (Intersecting Lines)
b) a₁/a₂ = b₁/b₂ = c₁/c₂ (Coincident Lines)
c) a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (Parallel Lines)
(i) 5x - 4y + 8 = 0 and 7x + 6 - 9 = 0
a₁ = 5, b₁ = - 4, c₁ = 8
a₂ = 7, b₂ = 6, c₂ = - 9
a₁/a₂ = 5/7...(1)
b₁/b₂ = -4/6 = -2/3...(2)
From (1) and (2)
a₁/a₂ ≠ b₁/b₂
Therefore, they are intersecting lines at a point.
(ii) 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0
a₁ = 9, b₁ = 3, c₁ = 12
a₂ = 18, b₂ = 6, c₂ = 24
a₁/a₂ = 9/18 = 1/2...(1)
b₁/b₂ = 3/6 = 1/2...(2)
c₁/c₂ = 12/24 = 1/2...(3)
From (1), (2) and (3)
a₁/a₂ = b₁/b₂ = c₁/c₂= 1/2
Therefore, they are coincident lines.
(iii) 6x – 3y + 10 = 0 and 2x – y + 9 = 0
a₁ = 6, b₁ = - 3, c₁ = 10
a₂ = 2, b₂ = - 1, c₂ = 9
a₁/a₂ = 6/2 = 3...(1)
b₁/b₂ = - 3/(- 1 ) = 3...(2)
c₁/c₂ = 10/9...(3)
From (1), (2) and (3)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Therefore, they are parallel lines.
