Question

In the following exercises, perform the indicated operations

$\frac{3y}{y+2}-\frac{y+2}{y+8}$

Short answer

Simplified expression is$\frac{2y^2+20y-4}{(y+2)(y+8)}$

Step-by-step solution

Step 1. Given Information  

Expression:

$\frac{3y}{y+2}-\frac{y+2}{y+8}$

Step 2. Find LCD 

Denominator of first fraction is $\left(y+2\right)$

Denominator of the second fraction is $\left(y+8\right)$

here nothing is common between both denominator

So the LCD is $\left(y+2\right)\left(y+8\right)$

Step 3. Rewrite each rational expression as  an equivalent rational expression with the LCD   

$\frac{3y(y+8)}{(y+2)(y+8)}-\frac{(y+2)(y+2)}{(y+2)(y+8)}$

Step 4.Simplify the numerators.

$$\begin{gathered} \frac{3y^2+24y}{(y+2)(y+8)}-\frac{{(y+2)}^2}{(y+2)(y+8)} \\ =\frac{3y^2+24y}{(y+2)(y+8)}-\frac{y^2+4y+4}{(y+2)(y+8)} \end{gathered}$$

Step 5. Subtract numerators

$\frac{3y^2+24y-(y^2+4y+4)}{(y+2)(y+8)}$

Step 6. Simplify numerator

$$\begin{gathered} \frac{3y^2+24y-y^2-4y-4}{(y+2)(y+8)} \\ =\frac{2y^2+20y-4}{(y+2)(y+8)} \end{gathered}$$