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

The answer is False.

The given pair of linear equations

x - 2y - 8 = 0

and 5x - 10y - c = 0

On Comparing with ax + by + c = 0;

Here,

Here, a_{1} = 1, b_{1} = - 2, c_{1} = - 8;

And a_{2} = 5, b_{2} = - 10, c_{2} = - c;

a_{1} /a_{2} = 1/5

b_{1} /b_{2} = 1/5

c_{1} /c_{2} = 8/c

But if c = 40 (real value), then the ratio c_{1}/c_{2} becomes 1/5 and then the system of linear equations has an infinitely many solutions.

Hence, at c= 40, the system of linear equations does not have a unique solution.

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