Find the value of k for which the system of equations has a unique solution: $4 x-5 y=k$ $2 x-3 y=12$

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Find the value of k for which the system of equations has a unique solution:
$4 x-5 y=k$
$2 x-3 y=12$

Find the value of k for which the system of equations has a unique solution:
$4 x-5 y=k$
$2 x-3 y=12$

Solution:

The system of given equations are
$\begin{array}{l} 4 x-5 y=k \\ \Rightarrow 4 x-5 y-k=0 \quad \text {....(i) } \end{array}$
And, $2 x-3 y=12$
$\Rightarrow 2 x-3 y-12=0\dots \dots(ii)$
The given equations are of the following form:
$a_{1} x+b_{1} y+c_{1}=0, a_{2} x+b_{2} y+c_{2}=0$
Here, $a_{1}=4, b_{1}=-5, c_{1}=-k$ and $a_{2}=2, b_{2}=-3, c_{2}=-12$
We must have for a unique solution:
$\begin{array}{l} \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} \\ \text { i.e. }, \frac{4}{2} \neq \frac{-5}{-3} \\ \Rightarrow 2 \neq \frac{5}{3} \Rightarrow 6 \neq 5 \end{array}$
As a result, for all the real values of $\mathrm{k}$ , the system of given equations will have a unique solution.