Question

Are the following pair of linear equations consistent?

Justify your answer.

$2ax+by=a$ , $4ax+2by-2a=0$ ,$a,b\ne 0$

Answer 1

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 $2ax+by-a=0$ and $4ax+2by-2a=0$

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

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

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

Here, $\frac{a_1}{a_2}=\frac{{\displaystyle 2a}}{{\displaystyle 4a}}=\frac{1}{2}$, $\frac{b_1}{b_2}=\frac{b}{2b}=\frac{1}{2}$, $\frac{c_1}{c_2}=\frac{{\displaystyle -a}}{{\displaystyle -2a}}=\frac{1}{2}$

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

Therefore the condition for consistency of the pair of linear equations is satisfied.

Hence, the given pair of linear equations are consistent and have infinitely many solutions.