# How do you combine (6y+5)/(5y-25)-(y+2)/(y-5)?

Mar 28, 2017

See the entire solution process below:

#### Explanation:

To subtract these two fractions they must be over a common denominator. To create a common denominator multiply the fraction on the right by the appropriate form of $1$ to not change the value of the function but to create a common denominator:

$\frac{6 y + 5}{5 y - 25} - \left(\frac{5}{5} \times \frac{y + 2}{y - 5}\right) \to$

$\frac{6 y + 5}{5 y - 25} - \frac{5 \times \left(y + 2\right)}{5 \times \left(y - 5\right)} \to$

$\frac{6 y + 5}{5 y - 25} - \frac{\left(5 \times y\right) + \left(5 \times 2\right)}{\left(5 \times y\right) - \left(5 \times 5\right)} \to$

$\frac{6 y + 5}{5 y - 25} - \frac{5 y + 10}{5 y - 25}$

We can next subtract the numerators over the common denominator:

$\frac{\left(6 y + 5\right) - \left(5 y + 10\right)}{5 y - 25} \to$

$\frac{6 y + 5 - 5 y - 10}{5 y - 25} \to$

$\frac{6 y - 5 y + 5 - 10}{5 y - 25} \to$

$\frac{\left(6 - 5\right) y + \left(5 - 10\right)}{5 y - 25} \to$

$\frac{1 y - 5}{5 y - 25} \to$

$\frac{y - 5}{5 y - 25}$

We can now factor the denominator and cancel common terms:

$\frac{y - 5}{\left(5 \times y\right) - \left(5 \times 5\right)} \to$

$\frac{y - 5}{5 \left(y - 5\right)} \to$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{y - 5}}}}{5 \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(y - 5\right)}}}} \to$

$\frac{1}{5}$

However, from the original expression: $\left(5 y - 25\right) \ne 0$ and $\left(y - 5\right) \ne 0$ therefore, $y \ne 5$

Mar 28, 2017

$\frac{1}{y}$

#### Explanation:

$\frac{6 y + 5}{\textcolor{red}{\left(5 y - 25\right)}} - \frac{y + 2}{y - 5} \text{ } \leftarrow$ factorise

$= \frac{6 y + 5}{\textcolor{red}{5 \left(y - 5\right)}} - \frac{y + 2}{y - 5} \text{ } \leftarrow$ find the LCD

$= \frac{6 y + 5 - 5 \left(y + 2\right)}{5 \left(y - 5\right)} \text{ } \leftarrow$ make equivalent fractions

$= \frac{6 y + 5 - 5 y - 10}{5 \left(y - 5\right)} \text{ } \leftarrow$remove brackets

$= \frac{y - 5}{y \left(y - 5\right)} \text{ } \leftarrow$ collect like terms

$= \frac{\cancel{\left(y - 5\right)}}{y \cancel{\left(y - 5\right)}} \text{ } \leftarrow$ cancel like factors

$\frac{1}{y}$