Sure, let's solve the question step-by-step.
### Part A: Calculate the slope between the first two points (-3, 0) and (0, -1)
To calculate the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], we use the slope formula:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, the points are [tex]\((-3, 0)\)[/tex] and [tex]\((0, -1)\)[/tex].
[tex]\[
x_1 = -3, \quad y_1 = 0, \quad x_2 = 0, \quad y_2 = -1
\][/tex]
Now substitute these values into the slope formula:
[tex]\[
\text{slope}_A = \frac{-1 - 0}{0 - (-3)} = \frac{-1}{0 + 3} = \frac{-1}{3} = -\frac{1}{3}
\][/tex]
So, the slope between the points [tex]\((-3, 0)\)[/tex] and [tex]\((0, -1)\)[/tex] is [tex]\(-\frac{1}{3}\)[/tex].
### Part B: Calculate the slope between the next two points (0, -1) and (-7, 4)
Again, we use the slope formula with the points [tex]\((0, -1)\)[/tex] and [tex]\((-7, 4)\)[/tex].
[tex]\[
x_3 = 0, \quad y_3 = -1, \quad x_4 = -7, \quad y_4 = 4
\][/tex]
Now, substitute these values into the slope formula:
[tex]\[
\text{slope}_B = \frac{4 - (-1)}{-7 - 0} = \frac{4 + 1}{-7 - 0} = \frac{5}{-7} = -\frac{5}{7}
\][/tex]
So, the slope between the points [tex]\((0, -1)\)[/tex] and [tex]\((-7, 4)\)[/tex] is [tex]\(-\frac{5}{7}\)[/tex].
### Part C: What do the slopes from parts A and B tell you about the relationship between the points?
We have calculated two slopes:
- Slope between [tex]\((-3, 0)\)[/tex] and [tex]\((0, -1)\)[/tex] is [tex]\(-\frac{1}{3}\)[/tex].
- Slope between [tex]\((0, -1)\)[/tex] and [tex]\((-7, 4)\)[/tex] is [tex]\(-\frac{5}{7}\)[/tex].
To determine the relationship between the three points, let's compare these slopes.
The two slopes are:
- [tex]\(-\frac{1}{3}\)[/tex]
- [tex]\(-\frac{5}{7}\)[/tex]
Since [tex]\(-\frac{1}{3} \neq -\frac{5}{7}\)[/tex], the slopes are not equal.
Conclusion:
The slopes are not the same, indicating that the points [tex]\((-3, 0)\)[/tex], [tex]\((0, -1)\)[/tex], and [tex]\((-7, 4)\)[/tex] do not lie on the same line. Therefore, the points are not collinear.
### Summary
- Part A: The slope between [tex]\((-3, 0)\)[/tex] and [tex]\((0, -1)\)[/tex] is [tex]\(-\frac{1}{3}\)[/tex].
- Part B: The slope between [tex]\((0, -1)\)[/tex] and [tex]\((-7, 4)\)[/tex] is [tex]\(-\frac{5}{7}\)[/tex].
- Part C: The points are not collinear because the slopes are different.
