If `3^(x-6)=9` and `3^(x+y)=81`, then value of y is:

Options

A. -4
B. 2
C. 3
D. 4

Answer

To solve the equations \(3^{(x-6)} = 9\) and \(3^{(x+y)} = 81\) and find the value of \(y\), we can follow these steps:

### Step 1: Solve the first equation
We start with the equation:
\[ 3^{(x-6)} = 9 \]
We know that \(9\) can be expressed as \(3^2\). Therefore, we rewrite the equation:
\[ 3^{(x-6)} = 3^2 \]
Since the bases are the same, we can set the exponents equal to each other:
\[ x - 6 = 2 \]

### Step 2: Solve for \(x\)
Now, we solve for \(x\):
\[ x = 2 + 6 = 8 \]

### Step 3: Substitute \(x\) into the second equation
Next, we use the value of \(x\) in the second equation:
\[ 3^{(x+y)} = 81 \]
Substituting \(x = 8\) gives:
\[ 3^{(8+y)} = 81 \]
Again, we can express \(81\) as \(3^4\):
\[ 3^{(8+y)} = 3^4 \]

### Step 4: Set the exponents equal
Since the bases are the same, we set the exponents equal:
\[ 8 + y = 4 \]

### Step 5: Solve for \(y\)
Now, we solve for \(y\):
\[ y = 4 - 8 = -4 \]

### Final Answer
Thus, the value of \(y\) is:
\[ \boxed{-4} \]

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