If `15(2x^(2) - y^(2)) = 7xy`, find `x : y`, if `x` and `y` both are positive.

Answer

To solve the equation \( 15(2x^2 - y^2) = 7xy \) and find the ratio \( x : y \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 15(2x^2 - y^2) = 7xy \]
We can rearrange it to:
\[ 2x^2 - y^2 = \frac{7}{15}xy \]

### Step 2: Divide by \( xy \)
Next, we divide both sides by \( xy \):
\[ \frac{2x^2}{xy} - \frac{y^2}{xy} = \frac{7}{15} \]
This simplifies to:
\[ \frac{2x}{y} - \frac{y}{x} = \frac{7}{15} \]

### Step 3: Let \( \frac{x}{y} = a \)
Now, we let \( \frac{x}{y} = a \). Therefore, \( x = ay \). Substituting this into the equation gives:
\[ \frac{2(ay)}{y} - \frac{y}{ay} = \frac{7}{15} \]
This simplifies to:
\[ 2a - \frac{1}{a} = \frac{7}{15} \]

### Step 4: Multiply through by \( a \)
To eliminate the fraction, we multiply through by \( a \):
\[ 2a^2 - 1 = \frac{7}{15}a \]

### Step 5: Rearrange into a standard quadratic equation
Rearranging gives us:
\[ 2a^2 - \frac{7}{15}a - 1 = 0 \]
To eliminate the fraction, multiply the entire equation by 15:
\[ 30a^2 - 7a - 15 = 0 \]

### Step 6: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to \( 30 \times -15 = -450 \) and add to \( -7 \). The numbers are \( -25 \) and \( 18 \):
\[ 30a^2 - 25a + 18a - 15 = 0 \]
Grouping gives:
\[ 5a(6a - 5) + 3(6a - 5) = 0 \]
Factoring out \( (6a - 5) \):
\[ (6a - 5)(5a + 3) = 0 \]

### Step 7: Solve for \( a \)
Setting each factor to zero gives:
1. \( 6a - 5 = 0 \) → \( a = \frac{5}{6} \)
2. \( 5a + 3 = 0 \) → \( a = -\frac{3}{5} \) (not valid since \( x \) and \( y \) are positive)

### Step 8: Conclusion
Thus, we have:
\[ \frac{x}{y} = a = \frac{5}{6} \]
This means the ratio \( x : y = 5 : 6 \).