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Sagot :
To find the slope of the line passing through the points [tex]\((-3, 4)\)[/tex] and [tex]\(\left(3, \frac{5}{2}\right)\)[/tex], we follow these steps:
### Step-by-Step Solution:
1. Identify the coordinates of the points:
[tex]\[ (x_1, y_1) = (-3, 4) \][/tex]
[tex]\[ (x_2, y_2) = \left(3, \frac{5}{2}\right) \][/tex]
2. Use the slope formula:
The slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
3. Subtract the [tex]\(y\)[/tex]-coordinates:
[tex]\[ y_2 - y_1 = \frac{5}{2} - 4 = \frac{5}{2} - \frac{8}{2} = \frac{5 - 8}{2} = \frac{-3}{2} = -1.5 \][/tex]
The result is [tex]\(-1.5\)[/tex].
4. Subtract the [tex]\(x\)[/tex]-coordinates:
[tex]\[ x_2 - x_1 = 3 - (-3) = 3 + 3 = 6 \][/tex]
The result is [tex]\(6\)[/tex].
5. Calculate the slope:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1.5}{6} = -0.25 \][/tex]
Therefore, the slope of the line passing through the points [tex]\((-3, 4)\)[/tex] and [tex]\(\left(3, \frac{5}{2}\right)\)[/tex] is [tex]\(-0.25\)[/tex].
## Conclusion for Part C
### Part C: Relationship Between Points
In Part A and Part B, you calculated the slopes between different pairs of points. By comparing these slopes, you can determine the nature of their relationship. Specifically:
- If the slopes are equal, the points lie on the same line (they are collinear).
- If the slopes are different, the points do not lie on the same line.
Since you only provided the context for Part A and Part B but not their slopes, this is a general guidance:
- Compare the slope from the newly calculated slope with those from Part A and B.
- If all slopes are equal, the points lie on a straight line.
- If any slope differs, the points do not lie collinear.
In summary, the slope of the line connecting the points [tex]\((-3, 4)\)[/tex] and [tex]\(\left(3, \frac{5}{2}\right)\)[/tex] is [tex]\(-0.25\)[/tex], and this slope value will help you understand the relationship when compared to the slopes from other points.
### Step-by-Step Solution:
1. Identify the coordinates of the points:
[tex]\[ (x_1, y_1) = (-3, 4) \][/tex]
[tex]\[ (x_2, y_2) = \left(3, \frac{5}{2}\right) \][/tex]
2. Use the slope formula:
The slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
3. Subtract the [tex]\(y\)[/tex]-coordinates:
[tex]\[ y_2 - y_1 = \frac{5}{2} - 4 = \frac{5}{2} - \frac{8}{2} = \frac{5 - 8}{2} = \frac{-3}{2} = -1.5 \][/tex]
The result is [tex]\(-1.5\)[/tex].
4. Subtract the [tex]\(x\)[/tex]-coordinates:
[tex]\[ x_2 - x_1 = 3 - (-3) = 3 + 3 = 6 \][/tex]
The result is [tex]\(6\)[/tex].
5. Calculate the slope:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1.5}{6} = -0.25 \][/tex]
Therefore, the slope of the line passing through the points [tex]\((-3, 4)\)[/tex] and [tex]\(\left(3, \frac{5}{2}\right)\)[/tex] is [tex]\(-0.25\)[/tex].
## Conclusion for Part C
### Part C: Relationship Between Points
In Part A and Part B, you calculated the slopes between different pairs of points. By comparing these slopes, you can determine the nature of their relationship. Specifically:
- If the slopes are equal, the points lie on the same line (they are collinear).
- If the slopes are different, the points do not lie on the same line.
Since you only provided the context for Part A and Part B but not their slopes, this is a general guidance:
- Compare the slope from the newly calculated slope with those from Part A and B.
- If all slopes are equal, the points lie on a straight line.
- If any slope differs, the points do not lie collinear.
In summary, the slope of the line connecting the points [tex]\((-3, 4)\)[/tex] and [tex]\(\left(3, \frac{5}{2}\right)\)[/tex] is [tex]\(-0.25\)[/tex], and this slope value will help you understand the relationship when compared to the slopes from other points.
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