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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 are intersecting at one point.
Hence given pair of linear equations is consistent
Let as 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 that is E.Hence given pair of linear equation 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 have infinitely many solution.Hence, the given pair of linear equations is consistent Let us plot the graph of 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 is coincident.Therefore the above linear equations have infinitely many solutions.

Posted by

infoexpert27

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