# Are the following pair of linear equations consistent? Justify your answer.$\frac{3}{5} x-y=\frac{1}{2}$$\frac{1}{5} x-3 y=\frac{1}{6}$

Given :

The given pair of equations is,

$\frac{3}{5} x-y=\frac{1}{2}$

$\frac{1}{5} x-3 y=\frac{1}{6}$

To find :

We have to find whether the given pair of linear equations are consistent.

Solution:

We know that,

The condition for consistent pair of linear equations is,

$\frac{a_1}{a_2}≠\frac{b_1}{b_2}$                [For unique solution]

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$            [For infinitely many solutions]

$\frac{3}{5} x-y=\frac{1}{2}$

$10(\frac{3}{5}x)-10(y)=10(\frac{1}{2})$

$6x-10y-5=0$

$\frac{1}{5} x-3 y=\frac{1}{6}$

$30(\frac{1}{5}x)-30(3y)=30(\frac{1}{6})$

$6x-90y-5=0$

Here,

$a_1=6, b_1=-10, c_1=-5$

$a_2=6, b_2=-90, c_2=-5$

Therefore,

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

$\frac{b_1}{b_2}=\frac{-10}{-90}=\frac{1}{9}$

$\frac{c_1}{c_2}=\frac{-5}{-5}=1$

Here,

$\frac{a_1}{a_2}≠\frac{b_1}{b_2}$

Hence, the given pair of linear equations has unique solution and therefore consistent.

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Updated on: 10-Oct-2022

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