Do the following pair of linear equations have no solution? Justify your answer.
$ 3 x+y-3=0 $
$ 2 x+\frac{2}{3} y=2 $

Given :

The given pair of equations is,

\( 3 x+y-3=0 \)

\( 2 x+\frac{2}{3} y=2 \)

To find :

We have to find whether the given pair of equations has no solution.

Solution:

We know that,

The condition for no solution is

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

\( 3 x+y-3=0 \)

\( 3(2 x)+3(\frac{2}{3} y)=3(2) \)

$6x+2y-6=0$

Here,

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

$a_2=6, b_2=2, c_2=-6$

Therefore,

$\frac{a_1}{a_2}=\frac{3}{6}=\frac{1}{2}$

$\frac{b_1}{b_2}=\frac{1}{2}$

$\frac{c_1}{c_2}=\frac{-3}{-6}=\frac{1}{2}$

Here,

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

Hence, the given pair of linear equations represent coincident lines. 

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Updated on: 10-Oct-2022

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