We notice that in \((x + y)(\frac{1}{x} + \frac{1}{y})\) ,\( (\frac{1}{x}+\frac{1}{y})\) has the same denominator after cross-multiplication as \((\frac{x}{y} + \frac{y}{x})\) has.
Keeping this in mind, we will first cross multiply and add both the fractions in the brackets.
\((\frac{x}{y} + \frac{y}{x})\) = \(\frac{x^2 + y^2}{xy} \) = 5 (given)
\((\frac{1}{x} + \frac{1}{y})\) = \(\frac{x + y}{xy}\)
Now,
\((x + y)(\frac{1}{x} + \frac{1}{y})\) = \((x + y)\frac{x + y}{xy}\) =\( \frac{x^2 + y^2 + 2xy}{xy}\) = \(\frac{x^2 + y^2}{xy } + \frac{2xy}{xy}\) = \(5 + 2\) = \(7\)
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