Do the following equations represent a pair of coincident lines? Justify your answer.
$ -2 x-3 y=1 $
$ 6 y+4 x=-2 $

Given :

The given pair of equations is,

\( -2 x-3 y=1 \)
\( 6 y+4 x=-2 \)

To find :

We have to find whether the given pair of equations represent a pair of coincident lines.

Solution:

We know that,

The condition for coincident lines is

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

\( -2 x-3 y-1=0 \)
\( 6 y+4 x+2=0 \)

Here,

$a_1=-2, b_1=-3, c_1=-1$

$a_2=4, b_2=6, c_2=2$

Therefore,

$\frac{a_1}{a_2}=\frac{-2}{4}=\frac{-1}{2}$

$\frac{b_1}{b_2}=\frac{-3}{6}=\frac{-1}{2}$

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

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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