Which of the following pairs of linear equations are consistent inconsistent 2x y 6 0 4x 2y 4 0?

Given: Pair of linear equation: $2x + y - 6 = 0,\ 4x -2y-4=0$.

To do: To find whether the given pair of linear equations is consistent /inconsistent. If consistent, then obtain the solution graphically.

Solution:

$2x+y-6=0$

$4x-2y-4=0$

$2x+y=6\ \  ...( i)$

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

For equation $( i)$, 

$2x+y=6$

$\Rightarrow y=6-2x$

Plot point $( 0,\ 6)$ and $( 3,\ 0)$ on a graph and join then to get equation 

$3x+y=6$

For equation $( ii)$, 

$4x-2y=4$

$\Rightarrow y=\frac{4x-4}{2}$

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 the solution is consistent.

Updated on 10-Oct-2022 10:34:02

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Solution

Step 1. check pairs of linear equations are consistent/inconsistent.We have x+y=5,2x+2y=10Now, (adsbygoogle = window.adsbygoogle || []).push({}); a1a2=12b1b2=12 c1c2=12∴a1a2=b1b2=c 1c2Thus, the equations are coincident and they have infinite number of possible solutions.Step 2. Graphical solution.For,x+y=5 x 0 3 5 y 5 2 0 For,2x+2y=10 x 0 3 5 y 5 2 0 The graphical representation of the equations is:Here the lines are overlapping each other. Therefore, the equations have infinite possible solutions.Hence, the given equations are consistent and have infinitely many solutions.

Solution

The given equations are 2x + y - 6 = 0 and 4x - 2y - 4 = 0.Comparing these equations with a1x+b1y+c1=0 and a2x+b2y+c2=0, we get:a1=2,b1=1,and c1=−6 a2=4,b2=−2and c2=−4a1a2=24 = 12b1b2=1−2 = -12 andc1c2=−6−4=32Hence, a1a2≠b1b2 Therefore, these linear equations represent intersecting lines and thus have only one possible solution. Hence, the pair of linear equations is consistent.Graphical representation: 2x + y - 6 = 0 y = 6 - 2x x012y642 And, 4x - 2y -4 = 0 ⇒ y=4x−42 ⇒ y = 2x - 2 x123y024 Graphical representation From the figure, it can be observed that these lines are intersecting each other at the only one point i.e., (2,2).Thus, x = 2 and y = 2 is the solution for the given pair of equations.

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