To determine which ordered pairs could be points on a line parallel to the line that contains [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex], we need to check which pairs of points have the same slope as the line passing through these two points.
1. Calculate the slope of the line passing through [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex]:
The slope, [tex]\( m \)[/tex], of a line passing through 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]
Substituting the given points:
[tex]\[
m = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = \frac{2}{5}
\][/tex]
So, the slope [tex]\( m \)[/tex] is [tex]\( \frac{2}{5} \)[/tex].
2. Check each pair of points to see if they have the same slope:
- [tex]\((\mathbf{-2, -5})\)[/tex] and [tex]\((\mathbf{-7, -3})\)[/tex]:
[tex]\[
m = \frac{-3 - (-5)}{-7 - (-2)} = \frac{2}{-5} = -\frac{2}{5}
\][/tex]
This slope is not equal to [tex]\(\frac{2}{5}\)[/tex].
- [tex]\((\mathbf{-1, 1})\)[/tex] and [tex]\((\mathbf{-6, -1})\)[/tex]:
[tex]\[
m = \frac{-1 - 1}{-6 - (-1)} = \frac{-2}{-5} = \frac{2}{5}
\][/tex]
This slope is equal to [tex]\(\frac{2}{5}\)[/tex].
- [tex]\((\mathbf{0, 0})\)[/tex] and [tex]\((\mathbf{2, 5})\)[/tex]:
[tex]\[
m = \frac{5 - 0}{2 - 0} = \frac{5}{2}
\][/tex]
This slope is not equal to [tex]\(\frac{2}{5}\)[/tex].
- [tex]\((\mathbf{1, 0})\)[/tex] and [tex]\((\mathbf{6, 2})\)[/tex]:
[tex]\[
m = \frac{2 - 0}{6 - 1} = \frac{2}{5}
\][/tex]
This slope is equal to [tex]\(\frac{2}{5}\)[/tex].
- [tex]\((\mathbf{3, 0})\)[/tex] and [tex]\((\mathbf{8, 2})\)[/tex]:
[tex]\[
m = \frac{2 - 0}{8 - 3} = \frac{2}{5}
\][/tex]
This slope is equal to [tex]\(\frac{2}{5}\)[/tex].
3. Conclusion:
The pairs of points that have a slope equal to [tex]\(\frac{2}{5}\)[/tex], and thus could be on a line parallel to the line containing [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex], are:
[tex]\[
\boxed{(-1, 1) \text{ and } (-6, -1), (1, 0) \text{ and } (6, 2), (3, 0) \text{ and } (8, 2)}
\][/tex]
So, the ordered pairs that match are [tex]\((-1, 1) \text{ and } (-6, -1)\)[/tex], [tex]\((1, 0) \text{ and }(6, 2)\)[/tex], and [tex]\((3, 0) \text{ and } (8, 2)\)[/tex].
