I. ` 3p+2q - 58 = 0`
II.` 4q+4p= 92`

Options

A. if p is greater than q.
B. if p is smaller than q.
C. if p is equal q.
D. if p is either equal to or greater than q.

Explanation

To solve the given equations step by step, we will follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \( 3p + 2q - 58 = 0 \)
2. \( 4q + 4p = 92 \)

### Step 2: Rearrange the first equation
Let's rearrange the first equation to express it in a simpler form:
\[ 3p + 2q = 58 \]

### Step 3: Rearrange the second equation
Now, we can simplify the second equation:
\[ 4q + 4p = 92 \implies q + p = 23 \quad \text{(dividing the entire equation by 4)} \]
So, we have:
\[ p + q = 23 \]

### Step 4: Express q in terms of p
From the equation \( p + q = 23 \), we can express \( q \) in terms of \( p \):
\[ q = 23 - p \]

### Step 5: Substitute q in the first equation
Now, we substitute \( q \) in the first equation \( 3p + 2q = 58 \):
\[ 3p + 2(23 - p) = 58 \]

### Step 6: Simplify the equation
Expanding the equation:
\[ 3p + 46 - 2p = 58 \]
Now, combine like terms:
\[ p + 46 = 58 \]

### Step 7: Solve for p
Subtract 46 from both sides:
\[ p = 58 - 46 \implies p = 12 \]

### Step 8: Substitute p back to find q
Now that we have \( p \), we can find \( q \) using the equation \( q = 23 - p \):
\[ q = 23 - 12 \implies q = 11 \]

### Step 9: State the values of p and q
Thus, we have:
\[ p = 12 \quad \text{and} \quad q = 11 \]

### Step 10: Determine the relationship between p and q
Now we can state the relationship between \( p \) and \( q \):
\[ p > q \quad \text{(since 12 > 11)} \]

### Final Answer
The values are \( p = 12 \) and \( q = 11 \), and the relationship is \( p > q \).

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