If ` ( 2^(x)) ( 2 ^(y)) = 8 and (9^(x)) ( 3 ^(y))= 81 `, then (x, y) is :

Options

A. (1,2)
B. (2,1)
C. (1,1)
D. (2,2)

Answer

To solve the equations given in the problem, we will break it down step by step.

### Step 1: Rewrite the first equation
The first equation is given as:
\[ (2^x)(2^y) = 8 \]
We can simplify the left side using the property of exponents:
\[ 2^{x+y} = 8 \]
Since \(8\) can be expressed as \(2^3\), we can rewrite the equation as:
\[ 2^{x+y} = 2^3 \]

### Step 2: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ x + y = 3 \quad \text{(Equation 1)} \]

### Step 3: Rewrite the second equation
The second equation is:
\[ (9^x)(3^y) = 81 \]
We can express \(9\) as \(3^2\) and \(81\) as \(3^4\):
\[ (3^2)^x \cdot 3^y = 3^4 \]
This simplifies to:
\[ 3^{2x} \cdot 3^y = 3^4 \]
Using the property of exponents again, we combine the left side:
\[ 3^{2x + y} = 3^4 \]

### Step 4: Set the exponents equal to each other
Again, since the bases are the same, we can set the exponents equal:
\[ 2x + y = 4 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations
Now we have a system of two equations:
1. \(x + y = 3\)
2. \(2x + y = 4\)

We can solve this system using substitution or elimination. Let's use substitution. From Equation 1, we can express \(y\) in terms of \(x\):
\[ y = 3 - x \]

Now, substitute \(y\) in Equation 2:
\[ 2x + (3 - x) = 4 \]
Simplifying this gives:
\[ 2x + 3 - x = 4 \]
\[ x + 3 = 4 \]
\[ x = 4 - 3 \]
\[ x = 1 \]

### Step 6: Find \(y\)
Now that we have \(x\), we can find \(y\) using Equation 1:
\[ y = 3 - x = 3 - 1 = 2 \]

### Final Answer
Thus, the solution for \((x, y)\) is:
\[ (x, y) = (1, 2) \]