Question

Express the following as rational expression: $ \dfrac{x}{{x - y}} + \dfrac{y}{{x + y}} + \dfrac{{2xy}}{{{y^2} - {x^2}}} $ .

Answer 1

Hint: Here in this question, we have + symbol which represents the addition and we have to add the multiple rational numbers. We are going to simplify the sum of fractions by taking the LCM of the denominators of the fractions such that the value of the fractions remains unchanged and then add them up.

Complete step by step solution:
Here in this question, we have to add the rational numbers. As we know, the $ + $ sign indicates the addition. The numbers are in the form of fraction. In fraction we have 3 types: proper fraction, improper fraction and mixed fraction.
In the fraction the numerator is less than the denominator then it is a proper fraction. The numerator is greater than the denominator then it is an improper fraction. The fraction is a combination of the whole number and fraction then it is mixed fraction.
Here in this question, we have,
 $ \dfrac{x}{{x - y}} + \dfrac{y}{{x + y}} + \dfrac{{2xy}}{{{y^2} - {x^2}}} $
Taking the negative sign common from the last two terms, we get,
 $ \dfrac{x}{{x - y}} + \dfrac{y}{{x + y}} - \left( {\dfrac{{2xy}}{{{x^2} - {y^2}}}} \right) $
Now, the values of the denominator are not the same so we take LCM for the denominator.
Taking L.C.M. of denominators, we get $ \left( {{x^2} - {y^2}} \right) $ .
So, we have, $ \dfrac{x}{{x - y}} + \dfrac{y}{{x + y}} - \left( {\dfrac{{2xy}}{{{x^2} - {y^2}}}} \right) $
 $ \Rightarrow \dfrac{x}{{x - y}} \times \left( {\dfrac{{x + y}}{{x + y}}} \right) + \dfrac{y}{{x + y}} \times \left( {\dfrac{{x - y}}{{x - y}}} \right) - \left( {\dfrac{{2xy}}{{{x^2} - {y^2}}}} \right) $
 $ \Rightarrow \dfrac{{{x^2} + xy}}{{{x^2} - {y^2}}} + \dfrac{{xy - {y^2}}}{{{x^2} - {y^2}}} - \dfrac{{2xy}}{{{x^2} - {y^2}}} $
 $ \Rightarrow \dfrac{{{x^2} + xy + xy - {y^2} - 2xy}}{{{x^2} - {y^2}}} $
Cancelling the like terms with opposite signs, we get,
 $ \Rightarrow \dfrac{{{x^2} - {y^2}}}{{{x^2} - {y^2}}} $
Cancelling the common factors in numerator and denominator, we get,
 $ \Rightarrow 1 $
We can’t simplify further. Therefore, we have $ \dfrac{x}{{x - y}} + \dfrac{y}{{x + y}} + \dfrac{{2xy}}{{{y^2} - {x^2}}} = 1 $
So, the correct answer is “1”.

Note: While adding the two fractions we need to check the values of the denominator, if both denominators are having the same value then we can add the numerators. Suppose if the fractions have different denominators, we have to take LCM for the denominators and we simplify for further.