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

The given linear equations represent coincident lines, which means they are essentially the same line. Therefore, there are infinitely many solutions, and the system is consistent.

\(x + y = 5,\)
\(x = 5 − y\)

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

 

And 

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

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

 

The graphic representation shows overlapping lines, confirming infinite solutions and consistency.

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

The given linear equations represent parallel lines. Therefore, there are no common solutions, and the system 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,  \(\dfrac{a_1}{a_2} ≠  \dfrac{b_1}{b_2}\)

The given linear equations represent intersecting lines. Therefore, there is exactly one solution, and the system is consistent.

\(2x + y − 6 = 0\) , \(y = 6 − 2x\) ;

x	0	1	2
y	6	4	2

 

And 

\(4x − 2y − 4 = 0\) , \(y = 2x - 2\)

x	1	2	3
y	0	2	4

The graphic representation shows lines intersecting at the point (2, 2), which is the unique 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, \(\dfrac{a_1}{a_2}= \dfrac{b_1}{b_2} ≠ \dfrac{c_1}{c_2}\)

The given linear equations represent parallel lines. Therefore, there are no common solutions, and the system is inconsistent.