Condition for the pair of system to have unique solution

a₁/a₂ b₁/b₂

Let the equations are,

a₁x + b₁y + c₁ = 0

and a₂x + b₂y + c₂ = 0

Since, x = - 1 and y = 3 is the unique solution of these two equations, then

It must satisfy the equations -

a₁(-1) + b₁(3) + c₁ = 0

- a₁ + 3b₁ + c₁ = 0 …(i)

and a₂(- 1) + b₂(3) + c₂ = 0

- a₂ + 3b₂ + c₂ = 0 …(ii)

Since for the different values of a₁, b₁, c₁ and a₂, b₂, c₂ satisfy the Eqs. (i) and (ii).

Hence, infinitely many pairs of linear equations are possible.