Let's solve the problem step-by-step:
### Part A:
We need to choose two points from the table and calculate the slope between them.
First, let's choose the points (0, 4) and (-2, 2).
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates:
[tex]\[ (x_1, y_1) = (0, 4) \][/tex]
[tex]\[ (x_2, y_2) = (-2, 2) \][/tex]
Now plug in these values into the slope formula:
[tex]\[ \text{slope}_A = \frac{2 - 4}{-2 - 0} = \frac{-2}{-2} = 1.0 \][/tex]
Hence, the slope between the points (0, 4) and (-2, 2) is [tex]\(1.0\)[/tex].
### Part B:
Next, we'll choose two different points from the table and calculate the slope between them.
Let's choose the points (-2, 2) and (-4, 1).
Using the same slope formula:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the coordinates:
[tex]\[ (x_1, y_1) = (-2, 2) \][/tex]
[tex]\[ (x_2, y_2) = (-4, 1) \][/tex]
Now plug in these values into the slope formula:
[tex]\[ \text{slope}_B = \frac{1 - 2}{-4 - (-2)} = \frac{-1}{-2} = 0.5 \][/tex]
Hence, the slope between the points (-2, 2) and (-4, 1) is [tex]\(0.5\)[/tex].
### Part C:
Now we need to analyze what these slopes tell us about the relationship between the points.
From Part A, the slope [tex]\(\text{slope}_A\)[/tex] is [tex]\(1.0\)[/tex].
From Part B, the slope [tex]\(\text{slope}_B\)[/tex] is [tex]\(0.5\)[/tex].
Since the slopes are different ([tex]\(1.0 \neq 0.5\)[/tex]), the points are not collinear. This means the points do not lie on the same straight line. If the slopes were the same, it would indicate that the points lie on the same line and hence are collinear.
In conclusion, the slopes from parts A and B show that the points are not collinear and do not lie on the same line.
