To determine whether it matters which of the two points on a line you choose to call [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] when calculating the slope of the line, let's go through the steps of calculating the slope for a line segment defined by two points.
1. Given Two Points: Suppose you have two points: [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex].
2. Slope Formula: The slope [tex]\(m\)[/tex] of the line passing through these points can be found using the formula:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
3. Swapping Points: If we swap the points, the new points will be [tex]\( (x_2, y_2) \)[/tex] as the first point and [tex]\((x_1, y_1)\)[/tex] as the second point. Now the slope [tex]\(m'\)[/tex] using these points can be calculated as:
[tex]\[
m' = \frac{y_1 - y_2}{x_1 - x_2}
\][/tex]
4. Simplifying [tex]\(m'\)[/tex]:
Notice that:
[tex]\[
m' = \frac{y_1 - y_2}{x_1 - x_2} = \frac{-(y_2 - y_1)}{-(x_2 - x_1)} = \frac{y_2 - y_1}{x_2 - x_1} = m
\][/tex]
5. Conclusion: Both [tex]\(m\)[/tex] and [tex]\(m'\)[/tex] are calculated to be the same, showing that it does not matter if you designate the points as [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] or swap them.
Therefore, the answer to whether it matters which of the two points you choose to call [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
A. True
This result shows that the calculation of the slope is invariant to the order in which the points are designated.
