Find the value of k for which the following system of equations has a unique solution:
$5 x - 7 y - 5 = 0 2 x + k y - 1 = 0$

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Solution

The given system of equations is
$5 x - 7 y - 5 = 0 . . . . . (i) 2 x + k y - 1 = 0 . . . . . (i i)$
This system is of the form
$a_{1} x + b_{1} y + c_{1} = 0 a_{2} x + b_{2} y + c_{2} = 0$
where $a_{1} = 5, b_{1} = - 7, c_{1} = - 5 and a_{2} = 2, b_{2} = k, c_{2} = - 1$
For the given system of equations to have a unique solution, we must have
$\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \Rightarrow \frac{5}{2} \neq \frac{- 7}{k} \Rightarrow k \neq \frac{- 14}{5}$
Hence, $k \neq - \frac{14}{5}$ .

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