Question

Are the following pair of linear equations consistent?

Justify your answer.

$-3x-4y=12$ , $4y+3x=12$

Answer 1

Determine whether the given equations are consistent or not:

Given equations are $-3x-4y=12$ and $4y+3x=12$

A pair of linear equations $a_{1}x+b_1+c_1=0$ and $a_{2}x+b_2+c_2=0$ 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 $-3x-4y-12=0$ and $4y+3x-12=0$

Comparing these equations with $a_{1}x+b_1+c_1=0$ and $a_{2}x+b_2+c_2=0$ we get

$a_1=-3,b_1=-4,c_1=-12$

$a_2=3,b_2=4,c_2=-12$

Here, $\frac{a_1}{a_2}=\frac{-3}{3}=-1$, $\frac{b_1}{b_2}=\frac{-4}{4}=-1$, $\frac{c_1}{c_2}=\frac{-12}{-12}=1$

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

Therefore, 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.