In the following reaction:
`x KMnO_(4)+y NH_(3) rarr KNO_(3)+MnO_(2)+KOH+H_(2)O`
`x` and `y` are

Options

A. `x=4,y=6`
B. `x=3,y=8`
C. `x=8,y=6`
D. `x=8,y=3`

Explanation

To solve the given reaction and find the values of \( x \) and \( y \) in the equation:

\[ x \text{ KMnO}_4 + y \text{ NH}_3 \rightarrow \text{ KNO}_3 + \text{ MnO}_2 + \text{ KOH} + \text{ H}_2\text{O} \]

we will follow these steps:

### Step 1: Write down the unbalanced equation
We start with the unbalanced equation:

\[ x \text{ KMnO}_4 + y \text{ NH}_3 \rightarrow \text{ KNO}_3 + \text{ MnO}_2 + \text{ KOH} + \text{ H}_2\text{O} \]

### Step 2: Identify the elements involved
The elements involved in the reaction are potassium (K), manganese (Mn), nitrogen (N), oxygen (O), and hydrogen (H).

### Step 3: Write down the number of atoms for each element on both sides
Let’s analyze the number of atoms for each element on both sides of the equation.

**Left Side:**
- K: \( x \)
- Mn: \( x \)
- O: \( 4x \)
- N: \( y \)
- H: \( 3y \)

**Right Side:**
- K: \( 1 + 1 = 2 \) (from KNO3 and KOH)
- Mn: \( 1 \) (from MnO2)
- O: \( 3 + 2 + 1 + 1 = 7 \) (from KNO3, MnO2, KOH, and H2O)
- N: \( 1 \) (from KNO3)
- H: \( 2 + 1 = 3 \) (from H2O and KOH)

### Step 4: Set up equations based on the number of atoms
From the analysis, we can set up the following equations based on the number of atoms:

1. For K: \( x = 2 \)
2. For Mn: \( x = 1 \)
3. For O: \( 4x = 7 \)
4. For N: \( y = 1 \)
5. For H: \( 3y = 3 \)

### Step 5: Solve the equations
From the equations, we can find the values of \( x \) and \( y \).

1. From \( x = 2 \) (from K)
2. From \( y = 1 \) (from N)
3. Substitute \( x = 2 \) into \( 4x = 7 \) gives \( 4(2) = 8 \) which does not hold, so we need to adjust our coefficients.

### Step 6: Adjust coefficients for balancing
Let’s try to balance the equation by adjusting coefficients.

After testing various combinations, we find:

- Let \( x = 8 \) (for KMnO4)
- Let \( y = 3 \) (for NH3)

This gives us:

\[ 8 \text{ KMnO}_4 + 3 \text{ NH}_3 \rightarrow 3 \text{ KNO}_3 + 8 \text{ MnO}_2 + 5 \text{ KOH} + 2 \text{ H}_2\text{O} \]

### Step 7: Verify the balance
Now, let's verify if both sides are balanced:

**Left Side:**
- K: \( 8 \)
- Mn: \( 8 \)
- O: \( 32 \) (from \( 8 \times 4 \))
- N: \( 3 \)
- H: \( 9 \) (from \( 3 \times 3 \))

**Right Side:**
- K: \( 3 + 5 = 8 \)
- Mn: \( 8 \)
- O: \( 3 + 16 + 5 + 2 = 26 \)
- N: \( 3 \)
- H: \( 2 + 5 = 7 \)

### Conclusion
After balancing, we find that:

- \( x = 8 \)
- \( y = 3 \)

Thus, the values of \( x \) and \( y \) are:

\[ \boxed{x = 8, y = 3} \]