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)\(x + y = 5\)
\(2x + 2y = 10\)

\(\dfrac{a_1}{a_2} = \dfrac{1}{2} , \dfrac{b_1}{b_2} = \dfrac{1}{2}, \dfrac{c_1}{c_2} = \dfrac{5}{10} =\dfrac{1}{2}\)

Since \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}\)

Therefore, these linear equations are coincident pairs of lines and thus have an infinite number of possible solutions. Hence, the pair of linear equations is consistent. 

\(x + y = 5,\)
\(x = 5 − y\)

\(x\)	\(4\)	\(3\)	\(2\)
\(y\)	\(1\)	\(2\)	\(3\)

 

And 

\(2x + 2y =10\)
\(x= 10-\dfrac{2y}{2}\)

\(x\)	\(4\)	\(3\)	\(2\)
\(y\)	\(1\)	\(2\)	\(3\)

 

Hence, the graphic representation is as follows. 

From the figure, it can be observed that these lines are overlapping each other. 

Therefore, infinite solutions are possible for the given pair of equations.

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(ii) \(x − y = 8\)
\(3x − 3y = 16\)

\(\dfrac{a_1}{a_2} =\dfrac{1}{3} , \dfrac{b_1}{b_2}= \dfrac{-1}{-3} = \dfrac{1}{3}, \dfrac{c_1}{c_2} = \dfrac{8}{16} = \dfrac{1}{2}\)

Since,\(\dfrac{a_1}{a_2} =  \dfrac{b_1}{b_2} ≠ \dfrac{c_1}{c_2}\)

Therefore, these linear equations are parallel to each other and thus have no possible solution.

Hence, the pair of linear equations is inconsistent. 

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(iii) 2x + y − 6 = 0 
4x − 2y − 4 = 0

\(\dfrac{a_1}{a_2} = \dfrac{2}{4} =\dfrac{1}{2} , \dfrac{b_1}{b_2}= \dfrac{-1}{2} , \dfrac{c_1}{c_2} = \dfrac{-6}{-4} = \dfrac{3}{2}\)

Since,  \(\dfrac{a_1}{a_2} ≠  \dfrac{b_1}{b_2}\)

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution.

Hence, the pair of linear equations is consistent.

\(2x + y − 6 = 0\) , \(y = 6 − 2x\) ;

x	0	1	2
y	6	4	2

 

And 

\(4x − 2y − 4 = 0\) , \(y = 4x -\dfrac{4}{2}\)

x	1	2	3
y	0	2	4

Hence, the graphic representation is as follows. 

From the figure, it can be observed that these lines intersect each other at the only point i.e., (2, 2) and it is the solution for the given pair of equations.

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(iv)\(2x − 2y − 2 = 0\)
\(4x − 4y − 5 = 0\)

\(\dfrac{a_1}{a_2}= \dfrac{2}{4} = \dfrac{1}{2}, \dfrac{b_1}{b_2} = \dfrac{-2}{-4} = \dfrac{1}{2} , \dfrac{c_1}{c_2}=\dfrac{2}{5}\)

Since, \(\dfrac{a_1}{a_2}= \dfrac{b_1}{b_2} ≠ \dfrac{c_1}{c_2}\)

Therefore, these linear equations are parallel to each other and thus have no possible solution.

Hence, the pair of linear equations is inconsistent.