How do you add or subtract $(4xy)/(x^2-y^2) + ( x-y)/(x+y)$ ?

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Answer 1 · 2018-06-28T21:45:11

See a solution process below:

To add or subtract fractions they must be over a common denominator.

$x^2 - y^2 = (x + y)(x - y)$

Therefore we need to multiply the fraction on the right by $(x - y)(x - y)$ , a form of $1$ , to get both fractions over a common denominator:

$(4xy)/(x^2 - y^2) + ((x - y)/(x - y) xx (x - y)/(x + y)) =>$

$(4xy)/(x^2 - y^2) + ((x - y)(x - y))/((x - y)(x + y)) =>$

$(4xy)/(x^2 - y^2) + (x^2 - 2xy + y^2)/(x^2 - y^2)$

Now, we can add the numerators of the two fractions over their common denominator:

$(4xy + x^2 - 2xy + y^2)/(x^2 - y^2) =>$

$(x^2 + 4xy - 2xy + y^2)/(x^2 - y^2) =>$

$(x^2 + (4 - 2)xy + y^2)/(x^2 - y^2) =>$

$(x^2 + 2xy + y^2)/(x^2 - y^2) =>$

$((x + y)(x + y))/((x + y)(x - y)) =>$

$(color(red)(cancel(color(black)((x + y))))(x + y))/(color(red)(cancel(color(black)((x + y))))(x - y)) =>$

$(x + y)/(x - y)$

How do you add or subtract (4xy)/(x^2-y^2) + ( x-y)/(x+y)(4xy)/(x^2-y^2) + ( x-y)/(x+y)?