Find `(x)/(y)`, when `x^(2) + 6y^(2) = 5xy`.

Answer

To solve the equation \( x^2 + 6y^2 = 5xy \) and find \( \frac{x}{y} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^2 + 6y^2 = 5xy \]

### Step 2: Divide by \( y^2 \)
To simplify the equation, we divide every term by \( y^2 \):
\[ \frac{x^2}{y^2} + \frac{6y^2}{y^2} = \frac{5xy}{y^2} \]
This simplifies to:
\[ \frac{x^2}{y^2} + 6 = \frac{5x}{y} \]

### Step 3: Let \( t = \frac{x}{y} \)
Now, we can let \( t = \frac{x}{y} \). Therefore, \( \frac{x^2}{y^2} = t^2 \) and \( \frac{5x}{y} = 5t \). Substituting these into the equation gives:
\[ t^2 + 6 = 5t \]

### Step 4: Rearrange the equation
Rearranging the equation, we get:
\[ t^2 - 5t + 6 = 0 \]

### Step 5: Factor the quadratic equation
Now, we need to factor the quadratic equation:
\[ t^2 - 5t + 6 = (t - 2)(t - 3) = 0 \]

### Step 6: Solve for \( t \)
Setting each factor to zero gives us:
\[ t - 2 = 0 \quad \text{or} \quad t - 3 = 0 \]
Thus, we find:
\[ t = 2 \quad \text{or} \quad t = 3 \]

### Step 7: Conclusion
Since \( t = \frac{x}{y} \), we have:
\[ \frac{x}{y} = 2 \quad \text{or} \quad \frac{x}{y} = 3 \]

### Final Answer:
Therefore, the values of \( \frac{x}{y} \) are \( 2 \) and \( 3 \).

---