By the graphical method, find whether the following pair of equations are consistent or not. If consistent, solve them.
$ x-2 y=6 $
$ 3 x-6 y=0 $

Given:

Pair of linear equations:

\( x-2 y=6 \)

\( 3 x-6 y=0 \)

To do:

We have to find whether the given pair of linear equations is consistent /inconsistent. If consistent, then obtain the solution graphically.

Solution:

$x-2y-6=0$.........(i)

$3x-6y=0$.........(ii)

Here,

$a_{1}=1, b_{1}=-2, c_{1}=-6$

$a_{2}=3, b_{2}=-6, c_{2}=0$

$\frac{a_{1}}{a_{2}}=\frac{1}{3}$

$\frac{b_{1}}{b_{2}}=\frac{-2}{-6}$

$=\frac{1}{3}$

$\frac{c_{1}}{c_{2}}=\frac{-6}{0}$

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}≠\frac{c_{1}}{c_{2}}$

This implies,

The lines represented by the given equations are parallel.

Hence, the given pair of lines is inconsistent.