The point (24,3) on a standard (x,y) coordinate plane is halfway between points (z,2x+1) and (15z,z-4) . What is the value of z ?

Options

A. 1
B. 1.5
C. 3
D. 7

Answer

To solve the problem, we need to find the value of \( z \) given that the point \( (24, 3) \) is the midpoint between the points \( (z, 2x + 1) \) and \( (15z, z - 4) \).

### Step-by-Step Solution:

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

2. **Set Up the Coordinates**:
Here, the two points are:
- Point 1: \( (z, 2x + 1) \)
- Point 2: \( (15z, z - 4) \)

3. **Calculate the Midpoint**:
Using the midpoint formula, the x-coordinate of the midpoint is:
\[ \frac{z + 15z}{2} = \frac{16z}{2} = 8z \]
The y-coordinate of the midpoint is:
\[ \frac{(2x + 1) + (z - 4)}{2} = \frac{2x + z - 3}{2} \]

4. **Set Up the Equations**:
Since the midpoint is given as \( (24, 3) \), we can set up the following equations:
- For the x-coordinate:
\[ 8z = 24 \]
- For the y-coordinate:
\[ \frac{2x + z - 3}{2} = 3 \]

5. **Solve for \( z \)**:
From the x-coordinate equation:
\[ 8z = 24 \implies z = \frac{24}{8} = 3 \]

6. **Verify the y-coordinate**:
Substitute \( z = 3 \) into the y-coordinate equation:
\[ \frac{2x + 3 - 3}{2} = 3 \implies \frac{2x}{2} = 3 \implies x = 3 \]

7. **Conclusion**:
The value of \( z \) is \( \boxed{3} \).