Are the following pair of linear equations consistent? Justify your answer.
$ 2 a x+b y=a $
$ 4 a x+2 b y-2 a=0 ; \quad a, b
≠0 $

Given :

The given pair of equations is,

\( 2 a x+b y=a \)

\( 4 a x+2 b y-2 a=0 ;  a, b ≠ 0 \)

To find :

We have to find whether the given pair of linear equations are consistent.

Solution:

We know that,

The condition for consistent pair of linear equations is,

$\frac{a_1}{a_2}≠\frac{b_1}{b_2}$                [For unique solution]

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$            [For infinitely many solutions]

\( 2 a x+b y-a=0 \)

\( 4 a x+2 b y-2 a=0; a, b ≠ 0 \)

Here,

$a_1=2a, b_1=b, c_1=-a$

$a_2=4a, b_2=2b, c_2=-2a$

Therefore,

$\frac{a_1}{a_2}=\frac{2a}{4a}=\frac{1}{2}$

$\frac{b_1}{b_2}=\frac{b}{2b}=\frac{1}{2}$

$\frac{c_1}{c_2}=\frac{-a}{-2a}=\frac{1}{2}$

Here,

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Hence, the given pair of linear equations has infinitely many solutions and therefore consistent.   

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Updated on: 10-Oct-2022

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