If `2^(P)+3^(q)=17and2^(P+2)-3^(q+1)=5`, then find the value of p and q.

Options

A. `-2,3`
B. `2,-3`
C. 3, 2
D. 2, 3

Answer

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

### Step 1: Analyze the first equation
We start with the equation:
\[ 2^P + 3^Q = 17 \]
From this equation, we can express \(3^Q\) in terms of \(2^P\):
\[ 3^Q = 17 - 2^P \]

### Step 2: Substitute into the second equation
Now we take the second equation:
\[ 2^{P+2} - 3^{Q+1} = 5 \]
We can rewrite \(3^{Q+1}\) as \(3^Q \cdot 3\):
\[ 2^{P+2} - 3 \cdot 3^Q = 5 \]
Substituting \(3^Q\) from Step 1 into this equation gives:
\[ 2^{P+2} - 3(17 - 2^P) = 5 \]

### Step 3: Simplify the equation
Now we simplify the equation:
\[ 2^{P+2} - 51 + 3 \cdot 2^P = 5 \]
Rearranging gives:
\[ 2^{P+2} + 3 \cdot 2^P - 56 = 0 \]

### Step 4: Factor the equation
We can factor \(2^{P+2}\) as \(4 \cdot 2^P\):
\[ 4 \cdot 2^P + 3 \cdot 2^P - 56 = 0 \]
Combining like terms:
\[ (4 + 3) \cdot 2^P - 56 = 0 \]
This simplifies to:
\[ 7 \cdot 2^P = 56 \]

### Step 5: Solve for \(2^P\)
Dividing both sides by 7:
\[ 2^P = 8 \]
Thus, we find:
\[ P = 3 \quad (\text{since } 2^3 = 8) \]

### Step 6: Substitute \(P\) back to find \(Q\)
Now we substitute \(P = 3\) back into the first equation:
\[ 2^3 + 3^Q = 17 \]
This simplifies to:
\[ 8 + 3^Q = 17 \]
Subtracting 8 from both sides:
\[ 3^Q = 9 \]
Thus, we find:
\[ Q = 2 \quad (\text{since } 3^2 = 9) \]

### Final Answer
The values of \(P\) and \(Q\) are:
\[ P = 3, \quad Q = 2 \]