If `7x - 15y = 4x + y`, find the value of `x: y`. Hence, use componendo and dividendo to find the values of :
(i) `(9x + 5y)/(9x - 5y)`
(iI) `(3x^(2) + 2y^(2))/(3x^(2) - 2y^(2))`

Answer

To solve the equation \(7x - 15y = 4x + y\) and find the value of \(x : y\), we will follow these steps:

### Step 1: Rearranging the Equation
Start with the equation:
\[ 7x - 15y = 4x + y \]
Subtract \(4x\) from both sides:
\[ 7x - 4x - 15y = y \]
This simplifies to:
\[ 3x - 15y = y \]

### Step 2: Move \(y\) to the Left Side
Now, move \(y\) to the left side:
\[ 3x - 15y - y = 0 \]
This simplifies to:
\[ 3x - 16y = 0 \]

### Step 3: Express \(x\) in terms of \(y\)
Rearranging gives:
\[ 3x = 16y \]
Now, divide both sides by 3:
\[ x = \frac{16y}{3} \]

### Step 4: Finding the Ratio \(x : y\)
Now, to find \(x : y\), we can express it as:
\[ \frac{x}{y} = \frac{16}{3} \]
Thus, the ratio \(x : y\) is:
\[ x : y = 16 : 3 \]

### Step 5: Using Componendo and Dividendo
Now, we will use the ratio \(x : y = 16 : 3\) to find the values of the given expressions.

#### (i) Finding \(\frac{9x + 5y}{9x - 5y}\)
Using the ratio, we can write:
\[ \frac{9x + 5y}{9x - 5y} = \frac{9 \cdot \frac{16y}{3} + 5y}{9 \cdot \frac{16y}{3} - 5y} \]
This simplifies to:
\[ = \frac{\frac{144y}{3} + 5y}{\frac{144y}{3} - 5y} \]
Now, convert \(5y\) to a fraction with a denominator of 3:
\[ = \frac{\frac{144y + 15y}{3}}{\frac{144y - 15y}{3}} = \frac{159y/3}{129y/3} \]
The \(y/3\) cancels out:
\[ = \frac{159}{129} \]

### Step 6: Simplifying the Fraction
Now, simplify \(\frac{159}{129}\):
\[ = \frac{159 \div 3}{129 \div 3} = \frac{53}{43} \]

#### (ii) Finding \(\frac{3x^2 + 2y^2}{3x^2 - 2y^2}\)
Using the ratio again:
\[ \frac{3x^2 + 2y^2}{3x^2 - 2y^2} = \frac{3 \left(\frac{16y}{3}\right)^2 + 2y^2}{3 \left(\frac{16y}{3}\right)^2 - 2y^2} \]
Calculating \(x^2\):
\[ = \frac{3 \cdot \frac{256y^2}{9} + 2y^2}{3 \cdot \frac{256y^2}{9} - 2y^2} \]
This simplifies to:
\[ = \frac{\frac{768y^2}{9} + 2y^2}{\frac{768y^2}{9} - 2y^2} \]
Convert \(2y^2\) to a fraction with a denominator of 9:
\[ = \frac{\frac{768y^2 + 18y^2}{9}}{\frac{768y^2 - 18y^2}{9}} = \frac{786y^2/9}{750y^2/9} \]
The \(y^2/9\) cancels out:
\[ = \frac{786}{750} \]

### Step 7: Simplifying the Fraction
Now, simplify \(\frac{786}{750}\):
\[ = \frac{393}{375} \]

### Final Answers
1. The ratio \(x : y = 16 : 3\).
2. \(\frac{9x + 5y}{9x - 5y} = \frac{53}{43}\).
3. \(\frac{3x^2 + 2y^2}{3x^2 - 2y^2} = \frac{393}{375}\).