Question

Two first-degree equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 represent parallel lines if and only if a₁b₂ = a₂b₁ (i.e., a₁:b₁ = a₂:b₁(i.e , a₁:b₁ = a₂ : b₂).

Answer 1

When two lines are parallel, both are either vertical or non-vertical. If both are vertical, then b₁ = b₂ = 0 so that a₁b₂ = a₂b₁ = 0. If both are non-vertical, their slopes are equal.

Thus 

- a₁/b₁ = -a₂/b₂

Hence,   a₁b₂ = a₂b₁  or  a₁:b₁ = a₂:b₂ 

Conversely, when a₁b₂ = a₂b₁ and if b₁ = 0, then a₁ ≠ 0 so that b₂ = 0. Therefore, b₁ = 0 ⇔ b₂ = 0 and hence if one of the lines is vertical, then the other is also vertical so that the given lines are parallel. If both b₁ and b₂ are non-zero, then

 Since the slopes are equal, the lines are parallel.