Question

Are the following pair of linear equations consistent?

Justify your answer.

$x+3y=11$ , $2(2x+6y)=22$

Answer 1

Apply the condition for consistency of linear equations to determine whether the given equations are consistent or not:

A pair of linear equations $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ are said to be consistent if there exists a solution to these equations.

There can be a unique solution or infinitely many solutions to a pair of linear equations.

Mathematically it is represented by

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

The given pair of linear equations are $x+3y-11=0$ and $2x+6y-11=0$

Comparing these equations with $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ we get

$a_1=1,b_1=3,c_1=-11$

$a_2=2,b_2=6,c_1=-11$

Here, $\frac{a_1}{a_2}=\frac{1}{2}$, $\frac{b_1}{b_2}=\frac{3}{6}=\frac{1}{2}$, $\frac{c_1}{c_2}=\frac{{\displaystyle -11}}{{\displaystyle -11}}=1$

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

So, the condition for consistency of the pair of linear equations is not satisfied.

Hence, the given pair of linear equations are inconsistent and have no solution.