If the angle of inclination of line joining the points `(x,3)` and `(-2,5)` is `45^(@)`, find the value of `x`.

Answer

To solve the problem of finding the value of \( x \) such that the angle of inclination of the line joining the points \( (x, 3) \) and \( (-2, 5) \) is \( 45^\circ \), we can follow these steps:

### Step 1: Understand the Slope and Angle of Inclination
The slope \( m \) of a line that makes an angle \( \theta \) with the positive x-axis is given by:
\[ m = \tan(\theta) \]
For an angle of \( 45^\circ \):
\[ m = \tan(45^\circ) = 1 \]

### Step 2: Find the Slope of the Line Joining the Points
The slope \( m \) of the line joining the points \( (x, 3) \) and \( (-2, 5) \) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( (x_1, y_1) = (x, 3) \) and \( (x_2, y_2) = (-2, 5) \). Thus, we have:
\[ m = \frac{5 - 3}{-2 - x} = \frac{2}{-2 - x} \]

### Step 3: Set the Slope Equal to 1
Since we know the slope is equal to 1 (from the angle of inclination):
\[ \frac{2}{-2 - x} = 1 \]

### Step 4: Solve for \( x \)
To solve for \( x \), we can cross-multiply:
\[ 2 = -2 - x \]
Now, rearranging the equation gives:
\[ x = -2 - 2 \]
\[ x = -4 \]

### Step 5: Conclusion
The value of \( x \) is:
\[ \boxed{-4} \]