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Let us suppose that the cost price of TV and DVD be $x$ and $y$ rupees respectively.

Now, according to the first statement, it is given that if the shopkeeper sells a TV at $15\%$ profit and a DVD at $12\%$ loss then he earns Rs. 540 as total profit.

As we know that \[profit\%=\dfrac{profit}{CP}\times 100\%\Rightarrow profit=\dfrac{profit\%\times CP}{100}\]

Now, $15\%$profit on TV means $profit=\dfrac{15\times x}{100}=0.15x$

Similarly, we know that \[loss\%=\dfrac{loss}{CP}\times 100\%\Rightarrow loss=\dfrac{loss\%\times CP}{100}\]

Now, $12\%$ loss on DVD means $loss=\dfrac{12\times y}{100}=0.12y$

$\therefore $ Total profit = profit on TV – loss on DVD

$\Rightarrow 540=0.15x-0.12y...............(1)$

According to the second statement, it is given that if he sells the TV at a loss at $12\%$ loss and the DVD at $15\%$ profit then there is no profit or loss.

Now, $15\%$profit on DVD means $profit=\dfrac{15\times y}{100}=0.15y$

And, $12\%$ loss on TV means $loss=\dfrac{12\times x}{100}=0.12x$

$\therefore $ Net profit = profit on DVD – loss on TV

$\Rightarrow 0=0.15y-0.12x$

$\Rightarrow y=\dfrac{0.12}{0.15}x=\dfrac{4}{5}x................(2)$

Putting the value of $y$ in equation (1), we get

$\Rightarrow 540=0.15x-0.12\times \dfrac{4}{5}x$

$\Rightarrow 540=0.15x-0.096x$

$\Rightarrow 0.054x=540$

$\Rightarrow x=\dfrac{540}{0.054}=10000$

Now, putting the value of $x$ in equation (2), we get

$\Rightarrow y=\dfrac{4}{5}\times 10000=8000$

Hence, the value of $x$ and $y$ is 10000 and 8000 respectively.