Problem 77 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y^{2}-1}+\frac{2 y}{y-y^{2}}$$

Step-by-Step Solution

Answer

The solution to the task is \( -y/(y^{2}-1) \).

Step 1: Recognize the denominator

Recognize that \(y^{2}-1\) can be factored using the difference of squares to \( (y - 1)(y + 1) \).

Step 2: Adjust the second fraction

Rearrange the terms in the denominator of the second fraction to match the first, resulting in -\( (y - 1)(y + 1) \). Notice that, the second fraction is being divided by negative denominator, this calls for multiplication by -1 yielding \((-2y) /(y^{2}-1)\).

Step 3: Add the fractions

Now that the denominators of both fractions are identical, they can be added together. This results in \( (y - 2y) / (y^{2}-1) \).

Step 4: Simplify the fractions

Finally, combine the terms in the numerator, arriving at the final answer, \( -y/(y^{2}-1) \).