Question

if x+y /xy=7 and 3x+4y /xy=25, then what is the value of xy?

Answer 1

Dear student
$$\begin{gathered} \mathrm{Given}:\frac{\mathrm{x}+\mathrm{y}}{\mathrm{xy}}=7 \\ \Rightarrow \frac{1}{\mathrm{y}}+\frac{1}{\mathrm{x}}=7 \\ \frac{3\mathrm{x}+4\mathrm{y}}{\mathrm{xy}}=25 \\ \Rightarrow \frac{3}{\mathrm{y}}+\frac{4}{\mathrm{x}}=25 \\ \mathrm{Let}\frac{1}{\mathrm{x}}=\mathrm{m}\mathrm{and}\frac{1}{\mathrm{y}}=\mathrm{n} \\ \mathrm{n}+\mathrm{m}=7...\left(1\right)\times 3 \\ \Rightarrow 3\mathrm{n}+3\mathrm{m}=21...\left(2\right) \\ \mathrm{and}3\mathrm{n}+4\mathrm{m}=25...\left(3\right) \\ \mathrm{Subtracting}\left(3\right)\mathrm{from}\left(2\right) \\ -\mathrm{m}=-4 \\ \Rightarrow \mathrm{m}=4 \\ \mathrm{and}\mathrm{n}=3 \\ \Rightarrow \frac{1}{\mathrm{x}}=4\mathrm{and}\frac{1}{\mathrm{y}}=3 \\ \Rightarrow \mathrm{x}=\frac{1}{4}\mathrm{and}\mathrm{y}=\frac{1}{3} \\ \mathrm{So},\mathrm{xy}=\frac{1}{4}\times \frac{1}{3}=\frac{1}{12} \end{gathered}$$
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