If `(2)^(2x-2)-(8)^(y-1)=(16)^(x-2.5)`, then find the sum of x and y.

Options

A. 8
B. 7
C. 9
D. 6

Explanation

To solve the equation `(2)^(2x-2) - (8)^(y-1) = (16)^(x-2.5)`, we will first express all terms with the same base.

### Step-by-Step Solution:

1. **Rewrite the bases**:
- We know that \(8\) can be expressed as \(2^3\) and \(16\) can be expressed as \(2^4\).
- Therefore, we can rewrite the equation as:
\[ (2)^{2x-2} - (2^3)^{y-1} = (2^4)^{x-2.5} \]

2. **Apply the power of a power rule**:
- Using the rule \((a^m)^n = a^{m \cdot n}\), we can simplify the equation:
\[ (2)^{2x-2} - (2^{3(y-1)}) = (2^{4(x-2.5)}) \]
- This simplifies to:
\[ (2)^{2x-2} - (2^{3y-3}) = (2^{4x-10}) \]

3. **Set the equation with the same base**:
- Since the bases are the same, we can equate the exponents:
\[ 2x - 2 - (3y - 3) = 4x - 10 \]

4. **Simplify the equation**:
- Rearranging gives:
\[ 2x - 2 - 3y + 3 = 4x - 10 \]
- Combine like terms:
\[ 2x + 1 - 3y = 4x - 10 \]
- Rearranging further gives:
\[ -3y = 4x - 10 - 2x - 1 \]
\[ -3y = 2x - 11 \]
- Dividing by -3:
\[ y = \frac{11 - 2x}{3} \]

5. **Substituting back to find x**:
- Now, we will use another equation derived from the exponents:
\[ 2x - 2 = 3y - 3 \]
- Substitute \(y\) from the previous step:
\[ 2x - 2 = 3\left(\frac{11 - 2x}{3}\right) - 3 \]
- This simplifies to:
\[ 2x - 2 = 11 - 2x - 3 \]
\[ 2x - 2 = 8 - 2x \]
- Rearranging gives:
\[ 2x + 2x = 8 + 2 \]
\[ 4x = 10 \]
\[ x = \frac{10}{4} = 2.5 \]

6. **Finding y**:
- Substitute \(x = 2.5\) back into the equation for \(y\):
\[ y = \frac{11 - 2(2.5)}{3} = \frac{11 - 5}{3} = \frac{6}{3} = 2 \]

7. **Sum of x and y**:
- Now, calculate \(x + y\):
\[ x + y = 2.5 + 2 = 4.5 \]

### Final Answer:
The sum of \(x\) and \(y\) is \(4.5\).