Do the following equations represent a pair of coincident lines? Justify your answer.
$ \frac{x}{2}+y+\frac{2}{5}=0 $
$ 4 x+8 y+\frac{5}{16}=0 $

Given :

The given pair of equations is,

\(  \frac{x}{2}+y+\frac{2}{5}=0 \)

\( 4 x+8 y+\frac{5}{16}=0 \)

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,

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

\(  \frac{x}{2}+y+\frac{2}{5}=0 \)

$10(\frac{x}{2})+10(y)+10(\frac{2}{5})=0$

$5x+10y+4=0$

\( 4 x+8 y+\frac{5}{16}=0 \)

$16(4x)+16(8y)+16(\frac{5}{16})=0$

$64x+128y+5=0$

Here,

$a_1=5, b_1=10, c_1=4$

$a_2=64, b_2=128, c_2=5$

Therefore,

$\frac{a_1}{a_2}=\frac{5}{64}$

$\frac{b_1}{b_2}=\frac{10}{128}=\frac{5}{64}$

$\frac{c_1}{c_2}=\frac{4}{5}$

Here,

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

Hence, the given pair of linear equations has no solution.