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Simplify the expression:
[tex]\[ \frac{2x}{2x + 3y} + \frac{3y}{2x - 3y} - \frac{18y^2}{4x - 9y^2} \][/tex]


Sagot :

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]