By the graphical method, find whether the following pair of equations are consistent or not. If consistent, solve them.
$ 3 x+y+4=0 $
$ 6 x-2 y+4=0 $

To do:

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

Solution:

(i) $3x+y+4=0$.........(i)

$6x-2y+4=0$.........(ii)

Here,

$a_{1}=3, b_{1}=1, c_{1}=4$

$a_{2}=6, b_{2}=-2, c_{2}=4$

$\frac{a_{1}}{a_{2}}=\frac{3}{6}$

$=\frac{1}{2}$

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

$\frac{c_{1}}{c_{2}}=\frac{4}{4}$

$=1$

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

This implies,

The given pair of equations is consistent.

For equation (i),

$y=-4-3x$

Plot point $( 0,\ -4)$ and $(-1,\ -1)$ on a graph and join then to get equation. 

$3x+y+4=0$

For equation (ii), 

$2y=6x+4$

$y=3x+2$

Plot points $(-1,\ -1)$ and $( 0,\ 2)$ on a graph and join them to get equation $6x-2y+4=0$

$x=-1,\ y=-1$ is the solution of the given pairs of equation. So the solution is consistent.

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

(iii) $x+y-3=0$.........(i)

$3x+3y-9=0$.........(ii)

Here,

$a_{1}=1, b_{1}=1, c_{1}=-3$

$a_{2}=3, b_{2}=3, c_{2}=-9$

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

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

$\frac{c_{1}}{c_{2}}=\frac{-3}{-9}$

$=\frac{1}{3}$

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}=\frac{c_1}{c_2}$

This implies,

The given pair of lines is coincident and therefore consistent.

For equation (i),

$y=3-x$

Plot point $( 0,\ 3)$ and $(3,\ 0)$ on a graph and join then to get equation $x+y=3$

For equation (ii), 

$3y=9-3x$

$y=3-x$

Plot points $(0,\ 3)$ and $(3,\ 0)$ on a graph and join them to get equation $3x+3y=9$

There are infinite solutions as the lines are coincident.

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Updated on: 10-Oct-2022

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