Question

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) $2x-2y-2=0,4x-4y-5=0$

Answer 1

(i) $x+y=5$ ...(i)

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

$\Rightarrow x+y=5$

$\Rightarrow y=5-x$

x	0	3
y	5	2

Plot $(0,5)$ and $(3,5)$ on graph and join them to get equation $x+y=5$.

$2x+2y=10$

$\Rightarrow 2y=(10-2x)$

$\Rightarrow y=\frac{10-2x}{2}=5-x$ ...(iii)

x	5	2
y	0	3

So, the equation is consistent and has infinitely many solution

(ii) $x-y=8$ ....(i)

$3x-3y=16$ ....(ii)

$\Rightarrow x-y=8$

$\Rightarrow -x+y=-8$

$\Rightarrow y=-8+x$
$\Rightarrow y=x-8$

x	8	0
y	0	-8

$3x-3y=16$ ...(ii)

$\Rightarrow 3x=16+3y$

$\Rightarrow 3x-16=3y$

$\Rightarrow y=\frac{3x-16}{3}\Rightarrow y=x-\frac{16}{3}$

$x$	$\frac{16}{3}$	$0$
$y$	$0$	$\frac{-16}{3}$

Plotting both the equation in graph, we see that the lines are parallel , so inconsistent.

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

$4x-2y-4=0$

$2x+y=6$ ....(i)

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

For equation (i), $2x+y=6\Rightarrow y=6-2x$

$x$	$0$	$3$
$y$	$6$	$0$

Plot point $(0,6)$ and $(3,0)$ on a graph and join then to get equation $3x+y=6$

For equation (ii), $4x-2y=4\Rightarrow \frac{4x-4}{2}=y$

$x$	$1$	$0$
$y$	$0$	$-2$

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 pairs of equation . So. solution is consistent.

(iv) $2x-2y=2$ ...(i)

$4x-4y=5$ ...(ii)

$2x-2y=2\Rightarrow 2x-2=2y$

$y=x-1$

$x$	$0$	$1$
$y$	$-1$	$0$

Plot point $(0,1)$ and $(1,0)$ and join to get the equation $2x-2y=2$ on a graph $4x-4y=5\Rightarrow 4x-5=4y$

$\Rightarrow \frac{4x-5}{4}=y$

$x$	$0$	$\frac{5}{4}$
$y$	$-\frac{5}{4}$	$0$

Plot point $(0,-\frac{5}{4})$ and $(\frac{5}{4},0)$ and join them to get the equation $4x-4y=5$ on a graph.

The two lines never intersect, so, the solution is inconsistent.