Which of the following pairs of linear equations are consistent/inconsistent If consistent, obtain the solution graphically:
(i) $x + y = 5, 2x + 2y = 10$
(ii) $x – y = 8, 3x – 3y = 16$
(iii) $2x + y – 6 = 0, 4x – 2y – 4 = 0$
(iv) $2x – 2y – 2 = 0, 4x – 4y – 5 = 0$.

To do:

We have to find out whether the given pairs of linear equations are consistent or inconsistent and obtain the solution graphically.

Solution:

(i) $x+y-5=0$

$2x+2y-10=0$

$x+y=5\ \  ...( i)$

$2x+2y=10\ \ ...( ii)$

For equation $( i)$, 

$x+y=5$

$\Rightarrow y=5-x$

Plot point $( 0,\ 5)$ and $( 5,\ 0)$ on a graph and join then to get equation 

$x+y=5$

For equation $( ii)$, 

$2x+2y=10$

$\Rightarrow y=\frac{10-2x}{2}$

Plot point $( 5,\ 0)$ and $( 0,\ 5)$ on a graph and join them to get equation $2x+2y=0$

From the above figure, we can observe that the lines are coincident

Therefore, the equations have infinite possible solutions

Here, $a_1=1,\ b_1=-1,\ c_1=8$ and $a_2=3,\ b_2=-3,\ c_2=16$.

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

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

$\frac{c_1}{c_2}=\frac{8}{16}=\frac{1}{2}$

Here, we find that $\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}$

Thus, given pair of linear equations has no solution.

Therefore, the given pair of linear equations is inconsistent.

(iii) $2x+y-6=0$

$4x-2y-4=0$

$2x+y=6\ \  ...( i)$

$4x-2y=4\ \ ...( ii)$

For equation $( i)$, 

$2x+y=6$

$\Rightarrow y=6-2x$

x	1	0
y	0	-2

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

$x=2,\ y=2$ is the solution of the given pair of equations. So the solution is consistent.

(iv) Given equations: $2x – 2y – 2 = 0, 4x – 4y – 5 = 0$.

Here, $a_1=2,\ b_1=-2,\ c_1=-2$ and $a_2=4,\ b_2=-4,\ c_2=-5$.

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

$\frac{b_1}{b_2}=\frac{-2}{-4}=\frac{1}{2}$

$\frac{c_1}{c_2}=\frac{-2}{-5}=\frac{2}{5}$

Here, we find that $\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}$

Thus, given pair of linear equations has no solution.

Therefore, the given pair of linear equations is inconsistent.

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Updated on: 10-Oct-2022

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