Find the slope and inclination of the line through each pair of the following points:
(-1, -8) and (5, 7)

Answer

To find the slope and inclination of the line through the points (-1, -8) and (5, 7), we will follow these steps:

### Step 1: Identify the points
Let the two points be:
- Point 1: \( (x_1, y_1) = (-1, -8) \)
- Point 2: \( (x_2, y_2) = (5, 7) \)

### Step 2: Use the slope formula
The formula for the slope \( m \) of a line through two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values of the points into the formula:
\[ m = \frac{7 - (-8)}{5 - (-1)} \]

### Step 3: Simplify the expression
Calculating the numerator and denominator:
\[ m = \frac{7 + 8}{5 + 1} = \frac{15}{6} \]

### Step 4: Reduce the fraction
Now, we can simplify \( \frac{15}{6} \):
\[ m = \frac{15 \div 3}{6 \div 3} = \frac{5}{2} \]

### Step 5: Find the angle of inclination
The angle of inclination \( \theta \) can be found using the tangent function:
\[ \tan(\theta) = m \]
Thus:
\[ \tan(\theta) = \frac{5}{2} \]
To find \( \theta \), we take the arctangent (inverse tangent):
\[ \theta = \tan^{-1}\left(\frac{5}{2}\right) \]

### Final Results
- Slope \( m = \frac{5}{2} \)
- Angle of inclination \( \theta = \tan^{-1}\left(\frac{5}{2}\right) \)