If mid point coordinate of join (3,x) and `(-8,5)` is `((-5)/(2),7)` find x?

Answer

To find the value of \( x \) given the midpoint of the points \( (3, x) \) and \( (-8, 5) \) is \( \left(-\frac{5}{2}, 7\right) \), we can follow these steps:

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

### Step 2: Identify the Coordinates
Here, the coordinates of the points are:
- Point \( P = (3, x) \)
- Point \( Q = (-8, 5) \)
- Midpoint \( R = \left(-\frac{5}{2}, 7\right) \)

### Step 3: Set Up the Midpoint Equations
Using the midpoint formula, we can set up two equations based on the x-coordinates and y-coordinates:

1. For the x-coordinates:
\[ \frac{3 + (-8)}{2} = -\frac{5}{2} \]

2. For the y-coordinates:
\[ \frac{x + 5}{2} = 7 \]

### Step 4: Solve the x-coordinate Equation
First, let's solve the x-coordinate equation:
\[ \frac{3 - 8}{2} = -\frac{5}{2} \]
\[ \frac{-5}{2} = -\frac{5}{2} \]
This equation holds true, confirming our points are correct.

### Step 5: Solve the y-coordinate Equation
Now, let's solve the y-coordinate equation:
\[ \frac{x + 5}{2} = 7 \]
Multiply both sides by 2:
\[ x + 5 = 14 \]
Subtract 5 from both sides:
\[ x = 14 - 5 \]
\[ x = 9 \]

### Final Answer
Thus, the value of \( x \) is \( 9 \).

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