The mid-point of the segment joining (3m, 6) and (-4, 3n) is (1, 2m, -1). Find the values of m and n.

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) given the mid-point of the segment joining the points \( (3m, 6) \) and \( (-4, 3n) \) is \( (1, 2m - 1) \).

### Step 1: Write the Midpoint Formula
The midpoint \( M \) of a segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

### Step 2: Identify the Coordinates
Here, we have:
- Point A: \( (3m, 6) \)
- Point B: \( (-4, 3n) \)

### Step 3: Apply the Midpoint Formula
Using the midpoint formula, we can express the midpoint as:
\[ M = \left( \frac{3m + (-4)}{2}, \frac{6 + 3n}{2} \right) \]
This simplifies to:
\[ M = \left( \frac{3m - 4}{2}, \frac{6 + 3n}{2} \right) \]

### Step 4: Set the Midpoint Equal to Given Coordinates
We know that the midpoint is also given as \( (1, 2m - 1) \). Therefore, we can set up the following equations:
1. \( \frac{3m - 4}{2} = 1 \)
2. \( \frac{6 + 3n}{2} = 2m - 1 \)

### Step 5: Solve the First Equation for \( m \)
From the first equation:
\[ \frac{3m - 4}{2} = 1 \]
Multiply both sides by 2:
\[ 3m - 4 = 2 \]
Add 4 to both sides:
\[ 3m = 6 \]
Divide by 3:
\[ m = 2 \]

### Step 6: Substitute \( m \) into the Second Equation
Now substitute \( m = 2 \) into the second equation:
\[ \frac{6 + 3n}{2} = 2(2) - 1 \]
This simplifies to:
\[ \frac{6 + 3n}{2} = 4 - 1 \]
\[ \frac{6 + 3n}{2} = 3 \]
Multiply both sides by 2:
\[ 6 + 3n = 6 \]
Subtract 6 from both sides:
\[ 3n = 0 \]
Divide by 3:
\[ n = 0 \]

### Final Values
Thus, the values of \( m \) and \( n \) are:
\[ m = 2, \quad n = 0 \]