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.

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 $c$.

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Updated on: 10-Oct-2022

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