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Are the following pair of linear equations consistent? Justify your answer.
$ x+3 y=11 $
$ 2(2 x+6 y)=22 $
Given :
The given pair of equations is,
\( x+3 y=11 \)
\( 2(2 x+6 y)=22 \)
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]
\( x+3 y-11=0 \)
\( 4 x+12 y-22=0 \)
Here,
$a_1=1, b_1=3, c_1=-11$
$a_2=4, b_2=12, c_2=-22$
Therefore,
$\frac{a_1}{a_2}=\frac{1}{4}$
$\frac{b_1}{b_2}=\frac{3}{12}=\frac{1}{4}$
$\frac{c_1}{c_2}=\frac{-11}{-22}=\frac{1}{2}$
Here,
$\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}$
Hence, the given pair of linear equations has no solution and therefore inconsistent.