On comparing the ratios and 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) $\frac{3}{2}x + \frac{5}{3}y = 7; 9x – 10y = 14$
(iv) $5x-3y = 11; -10x + 6y = -22$
(v) $\frac{4x}{3} + 2y = 8; 2x + 3y = 12$.

To do:

We have to find out whether the given pairs of linear equations are consistent or inconsistent.

Solution:

(i) Given equations are: $3x + 2y=5;\ 2x – 3y=7$

$\frac{a_1}{a_2}=\frac{3}{2}$

$\frac{b_1}{b_2}=\frac{-2}{3}$

$\frac{c_1}{c_2}=\frac{5}{7}$

Here we find, $\frac{a_1}{a_2}≠\frac{b_1}{b_2}$

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

Therefore, the pair of linear equations is consistent.

(ii) Given equations are: $2x-3y=8;\ 4x-6y=9$

$\frac{a_1}{a_2}=\frac{2}{4}=\frac{1}{2}$

$\frac{b_1}{b_2}=\frac{-3}{-6}=\frac{1}{2}$

$\frac{c_1}{c_2}=\frac{8}{9}$

Here we find, $\frac{a_1}{a_2}=\frac{b_1}{b_2}≠\frac{c_1}{c_2}$

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

Thus, the pair of linear equations is inconsistent.

(iii) Given equations are: $\frac{3x}{2}+\frac{5y}{3}=7;\ 9x-10y=14$.

$\frac{a_1}{a_2}=\frac{\frac{3}{2}}{9}=\frac{1}{6}$

$\frac{b_1}{b_2}=\frac{\frac{5}{3}}{-10}=-\frac{1}{6}$

$\frac{c_1}{c_2}=\frac{7}{14}=\frac{1}{2}$

Here we find, $\frac{a_1}{a_2}≠\frac{b_1}{b_2}$

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

Therefore, the pair of linear equations is consistent.

(iv) Given equations are: $5x-3y=11;\ -10x+6y=-22$

$\frac{a_1}{a_2}=\frac{5}{-10}=-\frac{1}{2}$

$\frac{b_1}{b_2}=\frac{-3}{6}=-\frac{1}{2}$

$\frac{c_1}{c_2}=\frac{11}{-22}=-\frac{1}{2}$

Here we find, $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Therefore, these linear equations are coincident pairs of lines and thus have an infinite number of possible solutions.

Thus, the given pair of linear equations is consistent.

(v) Given equations are: $\frac{4x}{3}+2y=8;\ 2x+3y=12$

$\frac{a_1}{a_2}=\frac{\frac{4}{3}}{2}=\frac{2}{3}$

$\frac{b_1}{b_2}=\frac{2}{3}$

$\frac{c_1}{c_2}=\frac{8}{12}=\frac{2}{3}$

Here we find, $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Therefore, these linear equations are coincident pairs of lines and thus have an infinite number of possible solutions.

Therefore, the pair of linear equations is consistent.

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Updated on: 10-Oct-2022

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