If `(4x+3y)/(4x-3y)=7/4` use the properties to find the value of `(2x^2-11y^2)/(2x^2+11y^2)`

Answer

To solve the problem, we start with the equation given:

\[ \frac{4x + 3y}{4x - 3y} = \frac{7}{4} \]

### Step 1: Cross-multiply to eliminate the fraction

Cross-multiplying gives us:

\[ 4(4x + 3y) = 7(4x - 3y) \]

### Step 2: Expand both sides

Expanding both sides results in:

\[ 16x + 12y = 28x - 21y \]

### Step 3: Rearrange the equation

Rearranging the equation to isolate terms involving \(x\) on one side and terms involving \(y\) on the other side:

\[ 16x - 28x = -21y - 12y \]

This simplifies to:

\[ -12x = -33y \]

### Step 4: Simplify the equation

Dividing both sides by -3 gives:

\[ 4x = 11y \]

### Step 5: Express \(x\) in terms of \(y\)

From the equation \(4x = 11y\), we can express \(x\) in terms of \(y\):

\[ x = \frac{11}{4}y \]

### Step 6: Substitute \(x\) into the expression we need to evaluate

We need to evaluate:

\[ \frac{2x^2 - 11y^2}{2x^2 + 11y^2} \]

Substituting \(x = \frac{11}{4}y\):

\[ x^2 = \left(\frac{11}{4}y\right)^2 = \frac{121}{16}y^2 \]

### Step 7: Substitute \(x^2\) into the expression

Now substituting \(x^2\) into our expression:

\[ 2x^2 = 2 \cdot \frac{121}{16}y^2 = \frac{242}{16}y^2 \]

Now substituting into the expression:

\[ \frac{2x^2 - 11y^2}{2x^2 + 11y^2} = \frac{\frac{242}{16}y^2 - 11y^2}{\frac{242}{16}y^2 + 11y^2} \]

### Step 8: Convert \(11y^2\) to a fraction with a common denominator

Convert \(11y^2\) to have a denominator of 16:

\[ 11y^2 = \frac{176}{16}y^2 \]

### Step 9: Substitute and simplify

Now substituting this back into our expression:

\[ \frac{\frac{242}{16}y^2 - \frac{176}{16}y^2}{\frac{242}{16}y^2 + \frac{176}{16}y^2} = \frac{\frac{66}{16}y^2}{\frac{418}{16}y^2} \]

### Step 10: Simplify the fraction

The \(y^2\) terms cancel out:

\[ \frac{66}{418} \]

### Step 11: Simplify the fraction further

Dividing both the numerator and denominator by 2:

\[ \frac{33}{209} \]

Thus, the final value of \(\frac{2x^2 - 11y^2}{2x^2 + 11y^2}\) is:

\[ \frac{33}{209} \]