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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