Let's address each part of the question step-by-step:
### Part A: Calculate the Slope from [tex]$(-7,5)$[/tex] to [tex]$(-3,4)$[/tex]
The slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
For the points [tex]\((-7, 5)\)[/tex] and [tex]\((-3, 4)\)[/tex]:
- [tex]\(x_1 = -7\)[/tex]
- [tex]\(y_1 = 5\)[/tex]
- [tex]\(x_2 = -3\)[/tex]
- [tex]\(y_2 = 4\)[/tex]
Plug these values into the slope formula:
[tex]\[
\text{slope} = \frac{4 - 5}{-3 - (-7)} = \frac{4 - 5}{-3 + 7} = \frac{-1}{4} = -0.25
\][/tex]
So, the slope from [tex]\((-7, 5)\)[/tex] to [tex]\((-3, 4)\)[/tex] is [tex]\(-0.25\)[/tex].
### Part B: Calculate the Slope from [tex]$(-3,4)$[/tex] to [tex]$\left(3, \frac{5}{2}\right)$[/tex]
Given points are [tex]\((-3, 4)\)[/tex] and [tex]\(\left(3, \frac{5}{2}\right)\)[/tex]:
- [tex]\(x_2 = -3\)[/tex]
- [tex]\(y_2 = 4\)[/tex]
- [tex]\(x_3 = 3\)[/tex]
- [tex]\(y_3 = \frac{5}{2}\)[/tex]
Plug these values into the slope formula:
[tex]\[
\text{slope} = \frac{\frac{5}{2} - 4}{3 - (-3)} = \frac{\frac{5}{2} - 4}{3 + 3} = \frac{\frac{5}{2} - \frac{8}{2}}{6} = \frac{\frac{-3}{2}}{6} = \frac{-3}{2 \cdot 6} = \frac{-3}{12} = -\frac{1}{4} = -0.25
\][/tex]
So, the slope from [tex]\((-3, 4)\)[/tex] to [tex]\(\left(3, \frac{5}{2}\right)\)[/tex] is [tex]\(-0.25\)[/tex].
### Part C: Relationship Between All the Points
The slopes calculated in both Parts A and B are equal:
[tex]\[
-0.25 = -0.25
\][/tex]
Since the slopes between both pairs of points are equal, it indicates that the line segments between these points have the same rate of change. This tells us that the points [tex]\((-7, 5)\)[/tex], [tex]\((-3, 4)\)[/tex], and [tex]\(\left(3, \frac{5}{2}\right)\)[/tex] are collinear, meaning they all lie on the same straight line. This is because a constant slope throughout different segments between points indicates a single straight line passes through all of them.
So, the points are collinear.
