Find whether the following pair of linear equations is consistent or inconsistent x+3y=5 2x+6y=8

AcademicMathematicsNCERTClass 10

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.

Find whether the following pair of linear equations is consistent or inconsistent x+3y=5 2x+6y=8

Updated on 10-Oct-2022 13:19:43