Find the value for k which each of the following systems of equations has a unique solution:

$5 x - 7 y - 5 = 0, 2 x + k y - 1 = 0.$

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Solution

$5 x - 7 y - 5 = 0 2 x + k y - 1 = 0 a_{1} = 5, b_{1} = - 7, c_{1} = - 5 a_{2} = 2, b_{2} = k, c_{2} = - 1$

For a unique solution, we must have,

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

Now,

$\frac{5}{2} \neq \frac{- 7}{k} k \neq \frac{- 14}{5}$

Thus, for all real values of k other than $\frac{- 14}{5}$ , the given system of equations will have a unique solution.

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