If `(( 5 x - y))/( ( 5 x + y)) = (3)/( 7)` , what is the value of `(( 4 x ^(2) + y^(2) + 4xy))/( ( 9 x ^(2) + 16 y^(2) + 24 xy))`?

Options

A. `(16)/(121)`
B. `(3)/(7)`
C. `(18)/(49)`
D. `(1)/(6)`

Answer

To solve the equation \(\frac{5x - y}{5x + y} = \frac{3}{7}\) and find the value of \(\frac{4x^2 + y^2 + 4xy}{9x^2 + 16y^2 + 24xy}\), we will follow these steps:

### Step 1: Cross Multiply the Given Equation
We start with the equation:
\[ \frac{5x - y}{5x + y} = \frac{3}{7} \]
Cross multiplying gives us:
\[ 7(5x - y) = 3(5x + y) \]

### Step 2: Distribute Both Sides
Distributing both sides, we have:
\[ 35x - 7y = 15x + 3y \]

### Step 3: Rearrange the Equation
Now, we will rearrange the equation to isolate terms involving \(x\) and \(y\):
\[ 35x - 15x = 3y + 7y \]
This simplifies to:
\[ 20x = 10y \]

### Step 4: Simplify the Equation
Dividing both sides by 10 gives us:
\[ 2x = y \quad \text{or} \quad y = 2x \]

### Step 5: Substitute \(y\) into the Expression
Now we substitute \(y = 2x\) into the expression we need to evaluate:
\[ \frac{4x^2 + y^2 + 4xy}{9x^2 + 16y^2 + 24xy} \]
Substituting \(y\):
\[ = \frac{4x^2 + (2x)^2 + 4x(2x)}{9x^2 + 16(2x)^2 + 24x(2x)} \]

### Step 6: Simplify the Numerator
Calculating the numerator:
\[ = \frac{4x^2 + 4x^2 + 8x^2}{9x^2 + 64x^2 + 48x^2} \]
This simplifies to:
\[ = \frac{16x^2}{121x^2} \]

### Step 7: Simplify the Expression
Now we can simplify:
\[ = \frac{16}{121} \]

### Final Answer
Thus, the value of \(\frac{4x^2 + y^2 + 4xy}{9x^2 + 16y^2 + 24xy}\) is:
\[ \frac{16}{121} \]

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