Sure! Let's go through the solution step-by-step.
### Part A: Calculating the Slope Between Two Points
1. Choose Two Points:
- Let's choose the points [tex]\((-2, -5)\)[/tex] and [tex]\( (2, -1) \)[/tex].
2. Slope Formula:
- 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]
3. Plug in the Coordinates:
- For points [tex]\((-2, -5)\)[/tex] and [tex]\( (2, -1) \)[/tex]:
[tex]\[
x_1 = -2, \; y_1 = -5, \; x_2 = 2, \; y_2 = -1
\][/tex]
4. Calculate the Slope:
[tex]\[
m = \frac{-1 - (-5)}{2 - (-2)} = \frac{-1 + 5}{2 + 2} = \frac{4}{4} = 1.0
\][/tex]
So, the slope between the points [tex]\((-2, -5)\)[/tex] and [tex]\( (2, -1) \)[/tex] is [tex]\( 1.0 \)[/tex].
### Part B: Calculating the Slope Between Two Different Points
1. Choose Two Different Points:
- Let's choose the points [tex]\((2, -1)\)[/tex] and [tex]\( (3, 3) \)[/tex].
2. Slope Formula:
- We use the same slope formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
3. Plug in the Coordinates:
- For points [tex]\( (2, -1) \)[/tex] and [tex]\( (3, 3) \)[/tex]:
[tex]\[
x_1 = 2, \; y_1 = -1, \; x_2 = 3, \; y_2 = 3
\][/tex]
4. Calculate the Slope:
[tex]\[
m = \frac{3 - (-1)}{3 - 2} = \frac{3 + 1}{3 - 2} = \frac{4}{1} = 4.0
\][/tex]
So, the slope between the points [tex]\( (2, -1) \)[/tex] and [tex]\( (3, 3) \)[/tex] is [tex]\( 4.0 \)[/tex].
### Part C: Relationship Between the Points
- The slopes calculated from parts A and B are different: [tex]\( 1.0 \)[/tex] and [tex]\( 4.0 \)[/tex] respectively.
- This difference in slopes indicates that the lines connecting these points have different steepness and therefore, the points do not lie on a straight line. If all points were collinear (on the same straight line), the slopes between any two pairs of points would have been the same. The fact that the slopes are different means these three points do not lie on the same straight line.
Therefore, the points [tex]\((-2, -5)\)[/tex], [tex]\( (2, -1) \)[/tex], and [tex]\( (3, 3) \)[/tex] do not all lie on the same straight line.
