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.

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

98 Views

Kickstart Your Career

Get certified by completing the course

Get Started