Question

On comparing the ratios\( \dfrac{a_1}{a_2},\) \(\dfrac{b_1}{b_2}\), and \(\dfrac{c_1}{c_2}\), find out whether the following pair of linear equations are consistent, or inconsistent.(i) \(3x + 2y = 5 ; 2x – 3y = 7\) (ii) \(2x – 3y = 8 ; 4x – 6y = 9\) (iii) \(\dfrac{3}{2x} + \dfrac{5}{3y} =7\) ; \(9x – 10y = 14\) (iv) \(5x – 3y = 11 \) ;\( – 10x + 6y = –22\) (v)\( \dfrac{4}{3x} +2y =8; 2x + 3y = 12\)

Answer 1

(i)\( 3x + 2y = 5 , 2x − 3y = 7 \)

\(\dfrac{a_1}{a_2} =\dfrac{3}{2}, \dfrac{b_1}{b_2} =\dfrac{-2}{3}, \dfrac{c_1}{c_2} =\dfrac{5}{7}\)

\(\dfrac{a_1}{a_2}≠ \dfrac{b_1}{b_2}\);

The linear equations intersect at a single point, resulting in one unique solution. Thus, the pair of linear equations is consistent.

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(ii) \(2x − 3y = 8 \) \(4x − 6y = 9\)

\(\dfrac{a_1}{a_2} =\dfrac{2}{4}= \dfrac{1}{2} , \dfrac{b_1}{b_2} = \dfrac{-3}{-6} = \dfrac{1}{2} , \dfrac{c_1}{c_2} = \dfrac{8}{9}\)

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

 The linear equations are parallel and have no solution. Hence, the pair of linear equations is inconsistent.

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(iii) \(\dfrac{3}{2 x}\) +\(\dfrac{5}{3y} =7\)
\(9x -10y =14\)
\(\dfrac{a_1}{a_2} = \dfrac{3}{22/9} =\dfrac{1}{6}\) , \(\dfrac{b_1}{b_2} =\dfrac{5}{3/(-10)} =\)\(\dfrac{-1}{6}\)  , \(\dfrac{c_1}{c_2}\) =\(\dfrac{7}{14} = \dfrac{1}{2}\)

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

The linear equations intersect at a single point, resulting in one unique solution. Thus, the pair of linear equations is consistent.

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(iv) \( 5x − 3 y = 11 \)
\(− 10x + 6y = − 22\)

\(\dfrac{a_1}{a_2} = \dfrac{5}{-10} = \dfrac{-1}{2} , \dfrac{b_1}{b_2} = \dfrac{-3}{6} =\dfrac{-1}{2}, \dfrac{c_1}{c_2} = \dfrac{11}{-22}= \dfrac{-1}{2}\)

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

 The linear equations represent coincident lines, resulting in an infinite number of solutions. Hence, the pair of linear equations is consistent. 

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(v) \(\dfrac{4}{3x} +2y =8\)
\(2x +3y =12 \)

\(\dfrac{a_1}{a_2} = \dfrac{4}{3/2} = \dfrac{2}{3} , \dfrac{b_1}{b_2} =\dfrac{2}{3} , \dfrac{c_1}{c_2} =\dfrac{9}{12} =\dfrac{2}{3}\)

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

The linear equations represent coincident lines, resulting in an infinite number of solutions. Hence, the pair of linear equations is consistent.