Let's simplify the given mathematical expression step by step:
[tex]\[
\frac{2 x}{2 x+3 y} + \frac{3 y}{2 x-3 y} - \frac{18 y^2}{4 x-9 y^2}
\][/tex]
### Step 1: Combine the first two fractions
To combine the fractions, we need a common denominator for the first two terms:
[tex]\[
\frac{2 x}{2 x+3 y} + \frac{3 y}{2 x-3 y}
\][/tex]
Let's find a common denominator for these two fractions which will be:
[tex]\[
(2x + 3y)(2x - 3y)
\][/tex]
Now, rewrite each fraction with the common denominator:
[tex]\[
\frac{2 x (2 x - 3 y)}{(2 x + 3 y)(2 x - 3 y)} + \frac{3 y (2 x + 3 y)}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
### Step 2: Simplify the numerators
Expand the numerators:
[tex]\[
\frac{2 x (2 x - 3 y)}{(2 x + 3 y)(2 x - 3 y)} = \frac{4x^2 - 6xy}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
[tex]\[
\frac{3 y (2 x + 3 y)}{(2 x + 3 y)(2 x - 3 y)} = \frac{6xy + 9y^2}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
Now, combine the two fractions:
[tex]\[
\frac{4x^2 - 6xy + 6xy + 9y^2}{(2 x + 3 y)(2 x - 3 y)} = \frac{4x^2 + 9y^2}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
So, we have:
[tex]\[
\frac{4x^2 + 9y^2}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
### Step 3: Address the third fraction
Now we consider the third term:
[tex]\[
- \frac{18 y^2}{4 x-9 y^2}
\][/tex]
Notice that [tex]\(4 x - 9 y^2\)[/tex] is actually the expanded form of [tex]\((2 x + 3 y)(2 x - 3 y)\)[/tex].
So we can rewrite the fraction as:
[tex]\[
- \frac{18 y^2}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
### Step 4: Combine all the fractions
Now combine all the terms into a single fraction with common denominator [tex]\((2 x + 3 y)(2 x - 3 y)\)[/tex]:
[tex]\[
\frac{4 x^2 + 9 y^2}{(2 x + 3 y)(2 x - 3 y)} - \frac{18 y^2}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
Combine the numerators:
[tex]\[
\frac{4 x^2 + 9 y^2 - 18 y^2}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
Simplify the numerator:
[tex]\[
4 x^2 + 9 y^2 - 18 y^2 = 4 x^2 - 9 y^2
\][/tex]
So we now have:
[tex]\[
\frac{4 x^2 - 9 y^2}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
### Step 5: Simplify further
Noticing that [tex]\( 4 x^2 - 9 y^2 \)[/tex] can be factorized as [tex]\( (2 x - 3 y)(2 x + 3 y) \)[/tex], we have:
[tex]\[
\frac{(2 x - 3 y)(2 x + 3 y)}{(2 x + 3 y)(2 x - 3 y)}
\][/tex]
Since the numerator and denominator are identical (non-zero terms), they can be cancelled out:
[tex]\[
1
\][/tex]
Therefore, the simplified expression is:
[tex]\[
1
\][/tex]
