If `2^(x) + 3^(y) = 17 and 2^(x+1)-3^(y+1)=5` then what is the value of x?

Options

A. 3
B. 2
C. 1
D. 0

Explanation

To solve the equations \(2^x + 3^y = 17\) and \(2^{x+1} - 3^{y+1} = 5\), we can follow these steps:

### Step 1: Rewrite the equations
We have the two equations:
1. \(2^x + 3^y = 17\) (Equation 1)
2. \(2^{x+1} - 3^{y+1} = 5\) (Equation 2)

### Step 2: Simplify Equation 2
We can rewrite Equation 2:
\[ 2^{x+1} = 2 \cdot 2^x \quad \text{and} \quad 3^{y+1} = 3 \cdot 3^y \]
Thus, Equation 2 becomes:
\[ 2 \cdot 2^x - 3 \cdot 3^y = 5 \]

### Step 3: Substitute \(2^x\) and \(3^y\)
Let \(a = 2^x\) and \(b = 3^y\). Then we can rewrite the equations as:
1. \(a + b = 17\) (Equation 1)
2. \(2a - 3b = 5\) (Equation 2)

### Step 4: Solve the system of equations
From Equation 1, we can express \(b\) in terms of \(a\):
\[ b = 17 - a \]

Now substitute \(b\) into Equation 2:
\[ 2a - 3(17 - a) = 5 \]
Expanding this gives:
\[ 2a - 51 + 3a = 5 \]
Combine like terms:
\[ 5a - 51 = 5 \]
Add 51 to both sides:
\[ 5a = 56 \]
Now divide by 5:
\[ a = \frac{56}{5} = 11.2 \]

### Step 5: Find \(b\)
Now substitute \(a\) back into the equation for \(b\):
\[ b = 17 - 11.2 = 5.8 \]

### Step 6: Find \(x\) and \(y\)
Recall that \(a = 2^x\) and \(b = 3^y\):
\[ 2^x = 11.2 \quad \text{and} \quad 3^y = 5.8 \]

To find \(x\):
\[ x = \log_2(11.2) \]

### Step 7: Calculate \(x\)
Using the change of base formula:
\[ x = \frac{\log_{10}(11.2)}{\log_{10}(2)} \]
Calculating this gives approximately:
\[ x \approx 3.5 \]

### Final Answer
Thus, the value of \(x\) is approximately \(3.5\).