p and q are positive numbers satisfying 3 p + 2 pq = 4 and 5 q + pq = 3 . Find the value of p

Options

A. `1 or - (9)/( 5)`
B. ` (1)/(2) or - (20)/(30)`
C. `1 or - (20)/(3)`
D. `(1)/(2) or - (9)/( 5)`

Answer

To solve the equations given in the problem, we will follow these steps:

### Step 1: Write down the equations
We have two equations based on the problem statement:
1. \( 3p + 2pq = 4 \) (Equation 1)
2. \( 5q + pq = 3 \) (Equation 2)

### Step 2: Rearrange Equation 1 to express \( q \) in terms of \( p \)
From Equation 1, we can isolate \( q \):
\[ 2pq = 4 - 3p \]
\[ q = \frac{4 - 3p}{2p} \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 3 into Equation 2
Now, we substitute \( q \) from Equation 3 into Equation 2:
\[ 5\left(\frac{4 - 3p}{2p}\right) + p\left(\frac{4 - 3p}{2p}\right) = 3 \]
This simplifies to:
\[ \frac{5(4 - 3p)}{2p} + \frac{p(4 - 3p)}{2p} = 3 \]
Combining the fractions:
\[ \frac{20 - 15p + 4 - 3p}{2p} = 3 \]
\[ \frac{24 - 18p}{2p} = 3 \]

### Step 4: Clear the fraction by multiplying both sides by \( 2p \)
\[ 24 - 18p = 6p \]

### Step 5: Rearrange the equation to solve for \( p \)
Bringing all terms involving \( p \) to one side:
\[ 24 = 6p + 18p \]
\[ 24 = 24p \]
\[ p = 1 \]

### Step 6: Verify the solution
Now we can substitute \( p = 1 \) back into Equation 3 to find \( q \):
\[ q = \frac{4 - 3(1)}{2(1)} = \frac{4 - 3}{2} = \frac{1}{2} \]

### Step 7: Check if \( p = 1 \) and \( q = \frac{1}{2} \) satisfy both original equations
1. For Equation 1:
\[ 3(1) + 2(1)(\frac{1}{2}) = 3 + 1 = 4 \quad \text{(True)} \]
2. For Equation 2:
\[ 5(\frac{1}{2}) + (1)(\frac{1}{2}) = \frac{5}{2} + \frac{1}{2} = 3 \quad \text{(True)} \]

Both equations are satisfied.

### Final Answer
The value of \( p \) is \( \boxed{1} \).

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