Question

Are the following pair of linear equations consistent? Justify your answer.

2ax + by = a, 4ax + 2by – 2a = 0; a, b ≠ 0

Answer 1

Conditions for pair of linear equations to be consistent are:

`a_1/a_2 ≠ b_1/b_2` ......[Unique solution]

`a_1/a_2 = b_1/b_2 = c_1/c_2`......[Coincident or infinitely many solutions]

Yes.

The given pair of linear equations 

2ax + by – a = 0 and 4ax + 2by – 2a = 0

Comparing the above equations with ax + by + c = 0

We get,

a₁ = 2a, b₁ = b, c₁ = – a

a₂ = 4a, b₂ = 2b, c₂ = – 2a

`a_1/a_2 = 1/2`

`b_1/b_2 = 1/2`

`c_1/c_2 = 1/2`

Here, `a_1/a_2 = b_1/b_2 = c_1/c_2`

Hence, the given pair of linear equations has infinitely many solutions, i.e., consistent