Question

Perform the subtraction.
StartFraction 5 x Over x squared minus y squared EndFraction
minusStartFraction 6 Over y minus x EndFraction
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Part 1
StartFraction 5 x Over x squared minus y squared EndFraction
minusStartFraction 6 Over y minus x EndFraction
equals
  
enter your response here
​(Simplify your answer. Type your answer in factored​ form.)

Answer 1

To perform the subtraction

\[
\frac{5x}{x^2 - y^2} - \frac{6}{y - x},
\]

first, we note that \( x^2 - y^2 \) can be factored using the difference of squares:

\[
x^2 - y^2 = (x - y)(x + y).
\]

Next, we rewrite the second fraction \(\frac{6}{y - x}\). Notice that \(y - x = -(x - y)\). Thus, we can rewrite \(\frac{6}{y - x}\) as:

\[
\frac{6}{y - x} = -\frac{6}{x - y}.
\]

Now we'll rewrite our expression:

\[
\frac{5x}{(x - y)(x + y)} - \left(-\frac{6}{x - y}\right) = \frac{5x}{(x - y)(x + y)} + \frac{6}{x - y}.
\]

To add these fractions, we need a common denominator. The common denominator is \((x - y)(x + y)\). We will rewrite the second fraction to have this common denominator:

\[
\frac{6}{x - y} = \frac{6(x + y)}{(x - y)(x + y)}.
\]

Now we can add the two fractions:

\[
\frac{5x + 6(x + y)}{(x - y)(x + y)}.
\]

Now we simplify the numerator:

\[
5x + 6(x + y) = 5x + 6x + 6y = 11x + 6y.
\]

Thus, we can write the expression as:

\[
\frac{11x + 6y}{(x - y)(x + y)}.
\]

Since \(11x + 6y\) cannot be factored further (as it has no common factors), the final simplified answer is:

\[
\boxed{\frac{11x + 6y}{(x - y)(x + y)}}.
\]