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

Answer 1

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)#
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Answer 2

To add or subtract the given expressions, we need to find a common denominator. The common denominator for (4xy)/(x^2-y^2) and (x-y)/(x+y) is (x^2-y^2)(x+y).

To add the expressions, we multiply the numerator and denominator of the first fraction by (x+y), and the numerator and denominator of the second fraction by (x^2-y^2):

[(4xy)(x+y)]/[(x^2-y^2)(x+y)] + [(x-y)(x^2-y^2)]/[(x^2-y^2)(x+y)]

Expanding the numerators, we get:

(4x^2y + 4xy^2 + x^3 - x^2y - xy^2 + y^3)/(x^3 + x^2y - x^2y - xy^2 + xy^2 - y^3)

Simplifying the numerator and denominator, we have:

(4x^2y + 4xy^2 + x^3 - xy^2 + y^3)/(x^3 - y^3)

Therefore, the simplified expression is:

(4x^2y + 4xy^2 + x^3 - xy^2 + y^3)/(x^3 - y^3)

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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