If `3 ^( x + y) = 81 and 81 ^( x - y) = 3,` then the value of `x/y` is:

Options

A. `(15)/(34)`
B. `(15)/(17)`
C. `(17)/(15)`
D. `(17)/(30)`

Answer

To solve the equations \(3^{(x + y)} = 81\) and \(81^{(x - y)} = 3\), we will follow these steps:

### Step 1: Rewrite the equations in terms of base 3
We know that \(81\) can be expressed as a power of \(3\):
\[ 81 = 3^4 \]
Thus, we can rewrite the first equation:
\[ 3^{(x + y)} = 3^4 \]
Since the bases are the same, we can equate the exponents:
\[ x + y = 4 \quad \text{(Equation 1)} \]

### Step 2: Rewrite the second equation
Now, we rewrite the second equation using the same base:
\[ 81^{(x - y)} = 3 \]
This can be rewritten as:
\[ (3^4)^{(x - y)} = 3^1 \]
Using the power of a power property, we simplify:
\[ 3^{4(x - y)} = 3^1 \]
Again, equating the exponents gives us:
\[ 4(x - y) = 1 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have the system of equations:
1. \(x + y = 4\)
2. \(4(x - y) = 1\)

From Equation 2, we can solve for \(x - y\):
\[ x - y = \frac{1}{4} \quad \text{(Equation 3)} \]

### Step 4: Add Equations 1 and 3
Now we add Equation 1 and Equation 3:
\[ (x + y) + (x - y) = 4 + \frac{1}{4} \]
This simplifies to:
\[ 2x = 4 + \frac{1}{4} \]
To combine the terms on the right, convert \(4\) to a fraction:
\[ 4 = \frac{16}{4} \]
Thus:
\[ 2x = \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \]
Now, divide both sides by \(2\):
\[ x = \frac{17}{8} \]

### Step 5: Substitute \(x\) back to find \(y\)
Now we substitute the value of \(x\) back into Equation 1 to find \(y\):
\[ \frac{17}{8} + y = 4 \]
Rearranging gives:
\[ y = 4 - \frac{17}{8} \]
Convert \(4\) to a fraction:
\[ 4 = \frac{32}{8} \]
Thus:
\[ y = \frac{32}{8} - \frac{17}{8} = \frac{15}{8} \]

### Step 6: Find the value of \(\frac{x}{y}\)
Now we can find \(\frac{x}{y}\):
\[ \frac{x}{y} = \frac{\frac{17}{8}}{\frac{15}{8}} = \frac{17}{15} \]

### Final Answer
The value of \(\frac{x}{y}\) is:
\[ \frac{x}{y} = \frac{17}{15} \]

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