If `5x + 9y = 5` and `125x^(3) + 729y^(3) = 120` then the vlaue of the product of `x` and `y` is

Options

A. `1/9`
B. `1/135`
C. 45
D. 135

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations:**
\[ 5x + 9y = 5 \quad \text{(Equation 1)} \]
\[ 125x^3 + 729y^3 = 120 \quad \text{(Equation 2)} \]

2. **Using the Identity for Cubes:**
We can use the identity for the sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
However, we can also express \(125x^3 + 729y^3\) in terms of \(5x\) and \(9y\).

Let \(a = 5x\) and \(b = 9y\). Then, we can rewrite Equation 2:
\[ 125x^3 + 729y^3 = (5x)^3 + (9y)^3 = a^3 + b^3 \]

Using the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

3. **Substituting \(a\) and \(b\):**
From Equation 1, we know:
\[ a + b = 5 \]

Therefore, we can substitute this into our identity:
\[ 125x^3 + 729y^3 = (5)(a^2 - ab + b^2) \]

4. **Calculating \(a^2 - ab + b^2\):**
We need to find \(a^2 - ab + b^2\). We can express \(a^2 + b^2\) as:
\[ a^2 + b^2 = (a + b)^2 - 2ab = 5^2 - 2ab = 25 - 2ab \]
Thus,
\[ a^2 - ab + b^2 = (a^2 + b^2) - ab = (25 - 2ab) - ab = 25 - 3ab \]

5. **Substituting back into Equation 2:**
Now substituting back into Equation 2:
\[ 125x^3 + 729y^3 = 5(25 - 3ab) \]
Setting this equal to 120:
\[ 5(25 - 3ab) = 120 \]

6. **Solving for \(ab\):**
Dividing both sides by 5:
\[ 25 - 3ab = 24 \]
Rearranging gives:
\[ 3ab = 1 \quad \Rightarrow \quad ab = \frac{1}{3} \]

7. **Finding the Product \(xy\):**
Since \(ab = 5x \cdot 9y = 45xy\), we have:
\[ 45xy = \frac{1}{3} \]
Dividing both sides by 45:
\[ xy = \frac{1}{3} \cdot \frac{1}{45} = \frac{1}{135} \]

Thus, the value of the product \(xy\) is:
\[ \boxed{\frac{1}{135}} \]