1 It `log_(3)x+log_(3)y=2+log_(3)2` and `log_(3)(x+y)=2` Then

Options

A. `x=1`, `y=8`
B. `x=9`, `y=3`
C. `x=3`, `y=6`
D. none of these

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Rewrite the first equation
We start with the equation:
\[ \log_3 x + \log_3 y = 2 + \log_3 2 \]
Using the property of logarithms that states \(\log_a b + \log_a c = \log_a (bc)\), we can combine the left side:
\[ \log_3 (xy) = 2 + \log_3 2 \]

### Step 2: Simplify the right side
We can rewrite the right side using the property of logarithms:
\[ 2 + \log_3 2 = \log_3 (3^2) + \log_3 2 = \log_3 (9 \cdot 2) = \log_3 18 \]
Thus, we have:
\[ \log_3 (xy) = \log_3 18 \]

### Step 3: Equate the arguments of the logarithms
Since the logarithms are equal, we can equate the arguments:
\[ xy = 18 \]

### Step 4: Use the second equation
Now we consider the second equation:
\[ \log_3 (x + y) = 2 \]
This implies:
\[ x + y = 3^2 = 9 \]

### Step 5: Solve the system of equations
Now we have a system of two equations:
1. \(xy = 18\)
2. \(x + y = 9\)

We can solve for \(x\) and \(y\) using these equations. Let's express \(y\) in terms of \(x\) from the second equation:
\[ y = 9 - x \]
Substituting this into the first equation:
\[ x(9 - x) = 18 \]
Expanding this gives:
\[ 9x - x^2 = 18 \]
Rearranging leads to:
\[ x^2 - 9x + 18 = 0 \]

### Step 6: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = -9\), and \(c = 18\):
\[ x = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 1 \cdot 18}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{9 \pm \sqrt{81 - 72}}{2} = \frac{9 \pm \sqrt{9}}{2} = \frac{9 \pm 3}{2} \]
This gives us two potential solutions:
\[ x = \frac{12}{2} = 6 \quad \text{and} \quad x = \frac{6}{2} = 3 \]

### Step 7: Find corresponding values of \(y\)
Using \(x + y = 9\):
- If \(x = 6\), then \(y = 9 - 6 = 3\).
- If \(x = 3\), then \(y = 9 - 3 = 6\).

Thus, the solutions are:
\[ (x, y) = (6, 3) \quad \text{or} \quad (3, 6) \]

### Final Answer
The values of \(x\) and \(y\) are \(3\) and \(6\).

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