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  • (i)By the graphical method, find whether the following pair of equations are consistent or not. If consistent, solve them. 3x + y + 4 = 0 ; 6x– 2y + 4 = 0

By the graphical method, find whether the following pair of equations are consistent or not. If consistent, solve them.

(i) 3x + y + 4 = 0 ; 6x– 2y + 4 = 0
(ii) x– 2y = 6 ; 3x – 6y = 0
(iii) x + y = 3 ; 3x + 3y = 9

Answers (1)

(i)Solution:
Given equations are    3x + y + 4 = 0
6x – 2y + 4 = 0
Here a1 = 3, b1 = 1, c1 = 4
a2 = 6, b2 = –2, c2 = 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}= \frac{1}{1}

\frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}
Hence the given pair of linear equations intersect at one point.
Hence given pair of linear equations is consistent
Let us plot the graph of given equations
In equation 3x + y + 4 = 0

x 0 -1 -2
y -4 -1 2

In equation 6x – 2y + 4 = 0

x 0 -1 -2
y 2 -1 -4


Here two lines AB and CD intersect at only one point which is E. Hence given pair of linear equations is consistent.
(ii)Solution:
Given equations are    x –2y –6 = 0
3x – 6y  = 0
Here a1 = 1, b1 = –2, c1 = –6
a2 = 3, b2 = –6, c2 = 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}

Here \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}

Here the given pair of linear equations is parallel. Therefore it has no solution Hence given pair of linear equations is inconsistent
(iii)Solution:
The given equations are          x + y – 3 = 0
3x +3y – 9 = 0
Here a1 = 1, b1 = 1, c1 = –3
a2 = 3, b2 = 3, c2 = –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}

\Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}}
Therefore the given pair of equations is coincident and has infinitely many solutions.Hence, the given pair of linear equations is consistent Let us plot the graph of the given equations
x + y = 3

x 0 1 2
y 3 2 1

3x + 3y = 9

x 0 1 2
y 3 2 1


Here lines AB and CD are coincident. Therefore the above linear equations have infinitely many solutions.

Posted by

infoexpert27

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