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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