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)**$3x + 2y = 5 , 2x − 3y = 7$

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

$\dfrac{a_1}{a_2}≠ \dfrac{b_1}{b_2}$;

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.

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

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

Since $\dfrac{a_1}{a_2}≠ \dfrac{b_1}{b_2}$;

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.

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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, $\dfrac{a_1}{a_2} =\dfrac{b_1}{b_2} =\dfrac{c_1}{c_2}$

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.

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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, $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$

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.