How do you simplify the expression $ \dfrac{{\dfrac{y}{x} - \dfrac{x}{y}}}{{\dfrac{{x + y}}{{xy}}}} $ ?

Hint: To solve this problem we should know about addition and subtraction of two fractional numbers. Also about the operation on algebraic variables.
Most important thing we will use to simplify any expression is algebraic formula .
That is,
  $ {x^2} - {y^2} = (x - y)(x + y) $

Complete step-by-step answer:
Step 1: solve the numerator part of the given expression:
  $ \dfrac{{\dfrac{y}{x} - \dfrac{x}{y}}}{{\dfrac{{x + y}}{{xy}}}} = \dfrac{{\dfrac{{{y^2} - {x^2}}}{{xy}}}}{{\dfrac{{x + y}}{{xy}}}} $
Step 2: apply algebraic formula to further simplify numerator:
  $ \dfrac{{\dfrac{{{y^2} - {x^2}}}{{xy}}}}{{\dfrac{{x + y}}{{xy}}}} = \dfrac{{\dfrac{{\left( {y - x} \right)(y + x)}}{{xy}}}}{{\dfrac{{x + y}}{{xy}}}} $
Step 3: further simply by it by dividing numerator from denominator:
  $ \dfrac{{\dfrac{{{y^2} - {x^2}}}{{xy}}}}{{\dfrac{{x + y}}{{xy}}}} = \dfrac{{\left( {y - x} \right)(y + x)}}{{xy}}\dfrac{{xy}}{{x + y}} $
  $ \dfrac{{\left( {y - x} \right)(y + x)}}{{xy}}\dfrac{{xy}}{{x + y}} = y - x $
Hence, simplify equation for $ \dfrac{{\dfrac{y}{x} - \dfrac{x}{y}}}{{\dfrac{{x + y}}{{xy}}}} = y - x $ .
So, the correct answer is “y - x”.

Note: Application of algebra is average word problem, mixture word problem, distance, rate, Time word problem and complex word problem. It is use in our daily life in calculating unknown measures and anything dimension. Algebraic formula used in solving trigonometry and other trigonometric equations.