Answer :
Sure, let’s go through each part step-by-step:
### Part A: Slope from (-8, 9) to (-4, 6)
To find the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], you use the slope formula:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((-8, 9)\)[/tex] and [tex]\((-4, 6)\)[/tex]:
- [tex]\(x_1 = -8\)[/tex]
- [tex]\(y_1 = 9\)[/tex]
- [tex]\(x_2 = -4\)[/tex]
- [tex]\(y_2 = 6\)[/tex]
Plugging these values into the formula:
[tex]\[ \text{slope} = \frac{6 - 9}{-4 - (-8)} = \frac{6 - 9}{-4 + 8} = \frac{-3}{4} = -0.75 \][/tex]
So, the slope from [tex]\((-8, 9)\)[/tex] to [tex]\((-4, 6)\)[/tex] is [tex]\(-0.75\)[/tex].
### Part B: Slope from (-4, 6) to [tex]\(\left(2, \frac{3}{2}\right)\)[/tex]
Using the same slope formula:
For the points [tex]\((-4, 6)\)[/tex] and [tex]\(\left(2, \frac{3}{2}\right)\)[/tex]:
- [tex]\(x_1 = -4\)[/tex]
- [tex]\(y_1 = 6\)[/tex]
- [tex]\(x_2 = 2\)[/tex]
- [tex]\(y_2 = \frac{3}{2} = 1.5\)[/tex]
Plugging these values into the formula:
[tex]\[ \text{slope} = \frac{1.5 - 6}{2 - (-4)} = \frac{1.5 - 6}{2 + 4} = \frac{-4.5}{6} = -0.75 \][/tex]
So, the slope from [tex]\((-4, 6)\)[/tex] to [tex]\(\left(2, 1.5\right)\)[/tex] is [tex]\(-0.75\)[/tex].
### Part C: Interpretation of the Slopes
We’ve calculated the slopes between:
- [tex]\((-8, 9)\)[/tex] and [tex]\((-4, 6)\)[/tex]
- [tex]\((-4, 6)\)[/tex] and [tex]\(\left(2, 1.5\right)\)[/tex]
Both slopes are [tex]\(-0.75\)[/tex].
When the slopes between different pairs of points are equal, it indicates that the points lie on the same straight line. Hence, the points [tex]\((-8, 9)\)[/tex], [tex]\((-4, 6)\)[/tex], and [tex]\(\left(2, 1.5\right)\)[/tex] are collinear. This means that there is a linear relationship between these points, and they can all be described by the same linear equation.
### Part A: Slope from (-8, 9) to (-4, 6)
To find the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], you use the slope formula:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((-8, 9)\)[/tex] and [tex]\((-4, 6)\)[/tex]:
- [tex]\(x_1 = -8\)[/tex]
- [tex]\(y_1 = 9\)[/tex]
- [tex]\(x_2 = -4\)[/tex]
- [tex]\(y_2 = 6\)[/tex]
Plugging these values into the formula:
[tex]\[ \text{slope} = \frac{6 - 9}{-4 - (-8)} = \frac{6 - 9}{-4 + 8} = \frac{-3}{4} = -0.75 \][/tex]
So, the slope from [tex]\((-8, 9)\)[/tex] to [tex]\((-4, 6)\)[/tex] is [tex]\(-0.75\)[/tex].
### Part B: Slope from (-4, 6) to [tex]\(\left(2, \frac{3}{2}\right)\)[/tex]
Using the same slope formula:
For the points [tex]\((-4, 6)\)[/tex] and [tex]\(\left(2, \frac{3}{2}\right)\)[/tex]:
- [tex]\(x_1 = -4\)[/tex]
- [tex]\(y_1 = 6\)[/tex]
- [tex]\(x_2 = 2\)[/tex]
- [tex]\(y_2 = \frac{3}{2} = 1.5\)[/tex]
Plugging these values into the formula:
[tex]\[ \text{slope} = \frac{1.5 - 6}{2 - (-4)} = \frac{1.5 - 6}{2 + 4} = \frac{-4.5}{6} = -0.75 \][/tex]
So, the slope from [tex]\((-4, 6)\)[/tex] to [tex]\(\left(2, 1.5\right)\)[/tex] is [tex]\(-0.75\)[/tex].
### Part C: Interpretation of the Slopes
We’ve calculated the slopes between:
- [tex]\((-8, 9)\)[/tex] and [tex]\((-4, 6)\)[/tex]
- [tex]\((-4, 6)\)[/tex] and [tex]\(\left(2, 1.5\right)\)[/tex]
Both slopes are [tex]\(-0.75\)[/tex].
When the slopes between different pairs of points are equal, it indicates that the points lie on the same straight line. Hence, the points [tex]\((-8, 9)\)[/tex], [tex]\((-4, 6)\)[/tex], and [tex]\(\left(2, 1.5\right)\)[/tex] are collinear. This means that there is a linear relationship between these points, and they can all be described by the same linear equation.
