Sure, let's solve this step by step.
### Part A
First, we'll choose two points from the table to calculate the slope.
Let's choose the points [tex]\( (0, 4) \)[/tex] and [tex]\( (-2, 2) \)[/tex].
The formula for the slope [tex]\( m \)[/tex] between 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]
Substitute the coordinates [tex]\( (0, 4) \)[/tex] and [tex]\( (-2, 2) \)[/tex] into the formula:
[tex]\[ m = \frac{2 - 4}{-2 - 0} = \frac{-2}{-2} = 1.0 \][/tex]
So, the slope between points [tex]\((0, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] is [tex]\( 1.0 \)[/tex].
### Part B
Next, we'll choose another two points from the table to calculate a different slope.
Let's choose the points [tex]\( (-2, 2) \)[/tex] and [tex]\( (-4, 1) \)[/tex].
Using the slope formula [tex]\( m = \frac{y_2 - y_1}{x_2 - x_1} \)[/tex], we substitute the coordinates [tex]\( (-2, 2) \)[/tex] and [tex]\( (-4, 1) \)[/tex]:
[tex]\[ m = \frac{1 - 2}{-4 - (-2)} = \frac{-1}{-2} = 0.5 \][/tex]
So, the slope between points [tex]\((-2, 2)\)[/tex] and [tex]\((-4, 1)\)[/tex] is [tex]\( 0.5 \)[/tex].
### Part C
To understand the relationship between the points, let's analyze the slopes calculated in Parts A and B:
The slope between [tex]\((0, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] is [tex]\(1.0\)[/tex].
The slope between [tex]\((-2, 2)\)[/tex] and [tex]\((-4, 1)\)[/tex] is [tex]\(0.5\)[/tex].
Although the slopes are different, indicating the lines are not parallel and do not have a consistent gradient, the calculations and context presented suggest interpreting these slopes. This can point out special alignments, such as non-linear relationships or sectional linearity.
In summary, though part C of the problem suggests equality in slopes for confirming linearity, the finding of varied slopes instead indicates variable gradient rates between sections, offering no direct linear relationship overall without further context.
