State if the following is true or false.

For all real values of c, the pair of equations x-2y=8, 5y-10y=c have a unique solution.

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True

False

answer is B.

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Detailed Solution

We are given equations as,
x-2y=8
5y-10y=c
Two lines

a x_{1} + {by}_{1} + c_{1} = 0

and

a x_{2} + {by}_{2} + c_{2} = 0

have a unique solution only if

\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}

.
Since, the equations has unique solution,

a 1 a 2 ≠ b 1 b 2

\Rightarrow \frac{1}{5} \neq \frac{- 2}{- 10} \neq \frac{8}{c}

But,

\frac{1}{5} = \frac{- 2}{- 10}

\Rightarrow \frac{1}{5} = \frac{1}{5}

It does not satisfy the condition.
Therefore, the statement is false.
The correct statement is that for all real values of c, the pair of equations, x-2y=8, 5y-10y=c does not have a unique solution.
Hence, option 2 is correct.

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