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Find the value of $k$ for which the following system of equations has a unique solution:
$4x\ –\ 5y\ =\ k$
$2x\ –\ 3y\ =\ 12$
Given: The given equation are $4x\ –\ 5y\ =\ k$; $2x\ –\ 3y\ =\ 12$
To do:  Find the value of $k$ for which the following system of equations having infinitely many solutions.
Solution:
The given system of equation is:
$4x\ –\ 5y\ =\ k$
$2x\ –\ 3y\ =\ 12$
The system of equation is of the form $a_{1} x+b_{1} y=c_{1}\ and\ a_{2} x+b_{2} y=c_{2}$
For unique solution there is a condition.
$\frac{a_{1}}{a_{2}} \ =\frac{b_{1}}{b_{2}} $
So, $\frac{4}{2} = \frac{5}{3} $
We can see that the condition of $\frac{a_{1}}{a_{2}} \ =\frac{b_{1}}{b_{2}}$ is violated here,
$\frac{4}{2} \
eq \frac{5}{3}$
eq \frac{5}{3}$
Since k doesn't affect the condition for the equation to have a single solution, k can take any real value.
Therefore, k can take any real value.
Updated on: 10-Oct-2022
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