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

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

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

Solution:

The system of equations given are
$\begin{array}{l} 5 x-7 y-5=0 \\ 2 x+k y-1=0 \end{array}$
This given 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 $a_{2}=2, b_{2}=k, c_{2}=-1$
So now, for the system of equations to have a unique solution, we must have:
$\begin{array}{l} \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \\ \Rightarrow \frac{5}{2} \neq \frac{-7}{k} \\ \Rightarrow \mathrm{k} \neq-\frac{14}{5} \\ \text { As a result, } \mathrm{k} \neq-\frac{14}{5} \end{array}$