Find the slope and inclination of the line through each pair of the following points:
(10, 4) and (-2, -2)

Answer

To find the slope and inclination of the line through the points (10, 4) and (-2, -2), we will follow these steps:

### Step 1: Identify the points
Let the points be:
- Point 1: \( (x_1, y_1) = (10, 4) \)
- Point 2: \( (x_2, y_2) = (-2, -2) \)

### Step 2: Use the slope formula
The formula for the slope \( m \) of a line through two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

### Step 3: Substitute the values
Substituting the coordinates of the points into the slope formula:
\[ m = \frac{-2 - 4}{-2 - 10} \]

### Step 4: Simplify the expression
Calculating the numerator and denominator:
- Numerator: \( -2 - 4 = -6 \)
- Denominator: \( -2 - 10 = -12 \)

Now substituting these values back into the slope formula:
\[ m = \frac{-6}{-12} = \frac{1}{2} \]

### Step 5: Find the angle of inclination
The angle of inclination \( \theta \) can be found using the formula:
\[ \theta = \tan^{-1}(m) \]
Substituting the value of \( m \):
\[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \]

### Step 6: Conclusion
Thus, the slope of the line is \( \frac{1}{2} \) and the angle of inclination \( \theta \) is \( \tan^{-1}\left(\frac{1}{2}\right) \).

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