The mid-point of (3p,4) and (-2, 2q) is (2,6). Find the value of p + q:

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) given that the midpoint of the points \( (3p, 4) \) and \( (-2, 2q) \) is \( (2, 6) \).

### Step-by-step Solution:

1. **Understanding the Midpoint Formula**:
The midpoint \( M \) of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( (x_1, y_1) = (3p, 4) \) and \( (x_2, y_2) = (-2, 2q) \).

2. **Setting Up the Equations**:
Since the midpoint is given as \( (2, 6) \), we can set up two equations based on the midpoint formula:
\[ \frac{3p + (-2)}{2} = 2 \quad \text{(1)} \]
\[ \frac{4 + 2q}{2} = 6 \quad \text{(2)} \]

3. **Solving Equation (1)**:
Start with the first equation:
\[ \frac{3p - 2}{2} = 2 \]
Multiply both sides by 2:
\[ 3p - 2 = 4 \]
Add 2 to both sides:
\[ 3p = 6 \]
Divide by 3:
\[ p = 2 \]

4. **Solving Equation (2)**:
Now, solve the second equation:
\[ \frac{4 + 2q}{2} = 6 \]
Multiply both sides by 2:
\[ 4 + 2q = 12 \]
Subtract 4 from both sides:
\[ 2q = 8 \]
Divide by 2:
\[ q = 4 \]

5. **Finding \( p + q \)**:
Now that we have \( p \) and \( q \):
\[ p + q = 2 + 4 = 6 \]

### Final Answer:
The value of \( p + q \) is \( 6 \).

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