If the mid-point of a segment joining `A((x)/(2),(y+1)/(2))` and B(x + 1,y- 3) is C (5, - 2), find x,y.

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) given that the midpoint \( C \) of the segment joining points \( A \) and \( B \) is \( (5, -2) \). The coordinates of points \( A \) and \( B \) are given as \( A\left(\frac{x}{2}, \frac{y+1}{2}\right) \) and \( B(x + 1, y - 3) \).

### Step-by-Step Solution:

1. **Use the Midpoint Formula**:
The midpoint \( C \) of a line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ C\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
For our points:
- \( A\left(\frac{x}{2}, \frac{y+1}{2}\right) \)
- \( B(x + 1, y - 3) \)
- \( C(5, -2) \)

2. **Set Up the Equations**:
From the midpoint formula, we can set up the following equations:
\[ \frac{\frac{x}{2} + (x + 1)}{2} = 5 \]
\[ \frac{\frac{y+1}{2} + (y - 3)}{2} = -2 \]

3. **Solve for \( x \)**:
Start with the first equation:
\[ \frac{\frac{x}{2} + (x + 1)}{2} = 5 \]
Multiply both sides by 2:
\[ \frac{x}{2} + (x + 1) = 10 \]
Combine the terms:
\[ \frac{x}{2} + x + 1 = 10 \]
Convert \( x \) to a fraction:
\[ \frac{x}{2} + \frac{2x}{2} + 1 = 10 \]
\[ \frac{3x}{2} + 1 = 10 \]
Subtract 1 from both sides:
\[ \frac{3x}{2} = 9 \]
Multiply by \( \frac{2}{3} \):
\[ x = 6 \]

4. **Solve for \( y \)**:
Now, use the second equation:
\[ \frac{\frac{y+1}{2} + (y - 3)}{2} = -2 \]
Multiply both sides by 2:
\[ \frac{y+1}{2} + (y - 3) = -4 \]
Combine the terms:
\[ \frac{y+1}{2} + y - 3 = -4 \]
Convert \( y \) to a fraction:
\[ \frac{y+1}{2} + \frac{2y}{2} - 3 = -4 \]
\[ \frac{y+1 + 2y - 6}{2} = -4 \]
Combine the terms:
\[ \frac{3y - 5}{2} = -4 \]
Multiply both sides by 2:
\[ 3y - 5 = -8 \]
Add 5 to both sides:
\[ 3y = -3 \]
Divide by 3:
\[ y = -1 \]

5. **Final Values**:
The values of \( x \) and \( y \) are:
\[ x = 6, \quad y = -1 \]