To simplify the given expression:
[tex]\[
\frac{6}{y+3} - \frac{6-y}{3}
\][/tex]
we follow these steps:
### Step 1: Find a common denominator
The denominators in the given fractions are [tex]\( y+3 \)[/tex] and 3. The common denominator for these denominators is [tex]\( 3(y+3) \)[/tex].
### Step 2: Rewrite each fraction with the common denominator
To rewrite each fraction with the common denominator, we adjust the numerators accordingly.
For [tex]\(\frac{6}{y+3}\)[/tex], the denominator needs to be multiplied by 3:
[tex]\[
\frac{6}{y+3} = \frac{6 \cdot 3}{(y+3) \cdot 3} = \frac{18}{3(y+3)}
\][/tex]
For [tex]\(\frac{6-y}{3}\)[/tex], the denominator needs to be multiplied by [tex]\(y+3\)[/tex]:
[tex]\[
\frac{6-y}{3} = \frac{(6-y)(y+3)}{3(y+3)}
\][/tex]
### Step 3: Combine the two fractions
Now, we combine the fractions with the common denominator [tex]\(3(y+3)\)[/tex]:
[tex]\[
\frac{18}{3(y+3)} - \frac{(6-y)(y+3)}{3(y+3)}
\][/tex]
### Step 4: Simplify the numerator
Combine the numerators over the same denominator:
[tex]\[
\frac{18 - (6-y)(y+3)}{3(y+3)}
\][/tex]
We expand and simplify the expression in the numerator:
[tex]\[
18 - ( (6-y)(y+3)) = 18 - (6y + 18 - y^2 - 3y)
\][/tex]
Simplify the terms inside the parenthesis:
[tex]\[
6y + 18 - y^2 - 3y = -y^2 + 3y + 18
\][/tex]
Substitute back into the simplification of the numerator:
[tex]\[
18 - (-y^2 + 3y + 18) = 18 + y^2 - 3y - 18
\][/tex]
Combine like terms:
[tex]\[
18 + y^2 - 3y - 18 = y^2 - 3y
\][/tex]
### Step 5: Write the simplified fraction
Now, the expression becomes:
[tex]\[
\frac{y^2 - 3y}{3(y+3)}
\][/tex]
Finally, factor the numerator:
[tex]\[
\frac{y(y-3)}{3(y+3)}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{y(y-3)}{3(y+3)}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{\frac{y(y-3)}{3(y+3)}}
\][/tex]
