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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.