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.