Question

Combine the following rational expressions. 2y/(y²-10y+24)-y/(y²-11y+30)

Answer 1

The combined expression is (2y - y) / (y² - 10y + 24) + (y - 0) / (y² - 11y + 30), which simplifies to y / (y² - 10y + 24).

Step-by-step explanation:

To combine the given rational expressions, we first find a common denominator, which is the product of the two denominators: (y² - 10y + 24)(y² - 11y + 30). Then, we express each term with this common denominator:

`\[ (2y)/(y² - 10y + 24) - (y)/(y² - 11y + 30) \]`

The first term's denominator is (y² - 10y + 24), and the second term's denominator is (y² - 11y + 30). The common denominator is their product: (y² - 10y + 24)(y² - 11y + 30).

Now, rewrite each term with the common denominator:

`\[ (2y \cdot (y² - 11y + 30))/((y² - 10y + 24)(y² - 11y + 30)) - (y \cdot (y² - 10y + 24))/((y² - 10y + 24)(y² - 11y + 30)) \]`

Combine the numerators:

`\[ (2y^2 - 22y + 60 - y^2 + 10y - 24)/((y² - 10y + 24)(y² - 11y + 30)) \]`

Simplify the numerator:

`\[ (y^2 - 12y + 36)/((y² - 10y + 24)(y² - 11y + 30)) \]`

Factor the numerator:

`((y - 6)(y - 6))/((y² - 10y + 24)(y² - 11y + 30)) \]`

Cancel out common factors in the numerator and denominator:

`\[ ((y - 6))/((y² - 10y + 24)) \]`

Therefore, the final simplified expression is y / (y² - 10y + 24).