For the following system of equation determine the value of k for which the given system of equation has a unique solution.
$2 x + k y = 1$
$5 x - 7 y = 5$

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Solution

The given system of equations is
$2 x + k y - 1 = 0$
$5 x - 7 y - 5 = 0$

It is of the form $a_{x} + b_{1} y + c_{1} = 0$
$a_{2} x + b_{2} y + c_{2} = 0$

where $a_{1} = 2, b_{1} = k, c_{1} = - 1$
and $a_{2} = 5, b_{2} = - 7$ and $c_{2} = - 5$

For a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ i.e., $\frac{2}{5} \neq \frac{k}{- 7} \Rightarrow k \neq \frac{- 14}{5}$

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

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