To solve this question, we will follow the instructions given carefully and provide a detailed, step-by-step solution.
### Part I:
First, we need to determine the correct formula for the slope, [tex]\( m \)[/tex].
The formula for the slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
So, the correct choice is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
### Part II:
Now we will use the slope formula to find the slope of the line passing through each set of points. We will also describe the nature of the line.
#### A. Points (5, 7) and (-4, -2):
1. Identify the coordinates:
- [tex]\( x_1 = 5 \)[/tex]
- [tex]\( y_1 = 7 \)[/tex]
- [tex]\( x_2 = -4 \)[/tex]
- [tex]\( y_2 = -2 \)[/tex]
2. Substitute these values into the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 7}{-4 - 5} \][/tex]
3. Calculate the numerator and the denominator:
- Numerator: [tex]\(-2 - 7 = -9\)[/tex]
- Denominator: [tex]\(-4 - 5 = -9\)[/tex]
4. Compute the slope:
[tex]\[ m = \frac{-9}{-9} = 1.0 \][/tex]
5. Describe the line:
- Since the slope is positive (1.0), the line is positive.
#### B. Points (1, 3) and (1, -10):
1. Identify the coordinates:
- [tex]\( x_1 = 1 \)[/tex]
- [tex]\( y_1 = 3 \)[/tex]
- [tex]\( x_2 = 1 \)[/tex]
- [tex]\( y_2 = -10 \)[/tex]
2. Substitute these values into the slope formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-10 - 3}{1 - 1} \][/tex]
3. Analyze the denominator:
- Since [tex]\( x_1 = x_2 \)[/tex], the denominator is [tex]\( 1 - 1 = 0 \)[/tex].
4. When the denominator is 0, the slope is undefined.
5. Describe the line:
- Since the slope is undefined, the line is vertical.
### Summary:
- For points (5, 7) and (-4, -2): The slope is [tex]\( 1.0 \)[/tex], and the line is positive.
- For points (1, 3) and (1, -10): The slope is undefined, and the line is vertical.
