If ` 5 x + 9 y = 5 and 125 x^(3) + 729 y^(3) = 120` then the value 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 equations given:

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

2. **Cube the First Equation:**
We will cube both sides of equation (1):
\[ (5x + 9y)^3 = 5^3 \]
This gives:
\[ (5x + 9y)^3 = 125 \quad \text{(3)} \]

3. **Use the Expansion Formula:**
We can use the formula for the cube of a sum:
\[ (A + B)^3 = A^3 + B^3 + 3AB(A + B) \]
Here, let \( A = 5x \) and \( B = 9y \). Then:
\[ (5x + 9y)^3 = (5x)^3 + (9y)^3 + 3(5x)(9y)(5x + 9y) \]
Substituting the values:
\[ (5x)^3 = 125x^3 \quad \text{and} \quad (9y)^3 = 729y^3 \]
Thus, we have:
\[ 125x^3 + 729y^3 + 3(5x)(9y)(5x + 9y) = 125 \]

4. **Substituting the Known Values:**
From equation (1), we know \( 5x + 9y = 5 \). Substituting this into our equation gives:
\[ 125x^3 + 729y^3 + 3(5x)(9y)(5) = 125 \]

5. **Simplifying the Equation:**
We can simplify the equation:
\[ 125x^3 + 729y^3 + 135xy \cdot 5 = 125 \]
This simplifies to:
\[ 125x^3 + 729y^3 + 675xy = 125 \quad \text{(4)} \]

6. **Substituting Equation (2) into (4):**
From equation (2), we know:
\[ 125x^3 + 729y^3 = 120 \]
Substitute this into equation (4):
\[ 120 + 675xy = 125 \]

7. **Solving for \( xy \):**
Rearranging gives:
\[ 675xy = 125 - 120 \]
\[ 675xy = 5 \]
Dividing both sides by 675:
\[ xy = \frac{5}{675} \]
Simplifying this fraction:
\[ xy = \frac{1}{135} \]

8. **Final Answer:**
The value of the product \( xy \) is:
\[ \boxed{\frac{1}{135}} \]

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