If ` ( x^(3) + 3 y ^(2) x)/( y^(3) + 3 x ^(2) y) = (35)/( 19) ` what is `(x)/( y) = `

Options

A. A)`(7)/(6)`
B. B)`(5)/(6)`
C. C)`(5)/(1)`
D. D)`(7)/(6)`

Answer

To solve the equation

\[ \frac{x^3 + 3y^2x}{y^3 + 3x^2y} = \frac{35}{19} \]

we will use the Componendo and Dividendo rule. Here are the steps to find \(\frac{x}{y}\):

### Step 1: Apply Componendo and Dividendo
According to the Componendo and Dividendo rule, we can add and subtract the numerator and denominator.

1. **Add the numerator and denominator**:
\[ x^3 + 3y^2x + y^3 + 3x^2y = (x+y)^3 \]

2. **Subtract the denominator from the numerator**:
\[ x^3 + 3y^2x - (y^3 + 3x^2y) = (x-y)^3 \]

Thus, we can rewrite the equation as:
\[ \frac{(x+y)^3}{(x-y)^3} = \frac{54}{16} \]

### Step 2: Simplify the Right Side
Now, simplify the right side:
\[ \frac{54}{16} = \frac{27}{8} \]

### Step 3: Take the Cube Root
Now we can take the cube root of both sides:
\[ \frac{x+y}{x-y} = \frac{3}{2} \]

### Step 4: Cross Multiply
Cross multiplying gives us:
\[ 2(x+y) = 3(x-y) \]

### Step 5: Expand and Rearrange
Expanding both sides:
\[ 2x + 2y = 3x - 3y \]

Rearranging the equation:
\[ 2y + 3y = 3x - 2x \]
\[ 5y = x \]

### Step 6: Solve for \(\frac{x}{y}\)
Now, we can express \(\frac{x}{y}\):
\[ \frac{x}{y} = \frac{5y}{y} = 5 \]

### Final Answer
Thus, the value of \(\frac{x}{y}\) is:
\[ \frac{x}{y} = 5 \]

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