# For all real values of $c$, the pair of equations$x-2 y=8$$5 x-10 y=c$have a unique solution. Justify whether it is true or false.

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Given :

The given pair of equations is,

$x-2 y=8$
$5 x-10 y=c$

To find :

We have to find whether for all real values of $c$, the given pair of equations have a unique solution.

Solution:

We know that,

The condition for a unique solution is,

$\frac{a_1}{a_2}≠\frac{b_1}{b_2}=\frac{c_1}{c_2}$

$x-2 y-8=0$

$5 x-10 y-c=0$

Here,

$a_1=1, b_1=-2, c_1=-8$

$a_2=5, b_2=-10, c_2=-c$

Therefore,

$\frac{a_1}{a_2}=\frac{1}{5}$

$\frac{b_1}{b_2}=\frac{-2}{-10}=\frac{1}{5}$

$\frac{c_1}{c_2}=\frac{-8}{-c}=\frac{8}{c}$

Here, for any value of $c$, $\frac{a_1}{a_2}=\frac{b_1}{b_2}$.

Hence, the system of linear equations never has a unique solution for any value of $c$.

Updated on 10-Oct-2022 13:27:13