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 find the slope of the line passing through [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex] first and then check which of the given pairs of points produce the same slope.
1. Find the slope of the line through [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex]:
The formula for the slope 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]
Plugging in [tex]\((x_1, y_1) = (3,4)\)[/tex] and [tex]\((x_2, y_2) = (-2,2)\)[/tex]:
[tex]\[
m = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = \frac{2}{5} = 0.4
\][/tex]
2. Check each pair of points to see if they have the same slope:
- For [tex]\((-2, -5)\)[/tex] and [tex]\((-7, -3)\)[/tex]:
[tex]\[
m = \frac{-3 - (-5)}{-7 - (-2)} = \frac{-3 + 5}{-7 + 2} = \frac{2}{-5} = -0.4
\][/tex]
This slope is [tex]\(-0.4\)[/tex], not [tex]\(0.4\)[/tex], so this pair is not parallel.
- For [tex]\((-1, 1)\)[/tex] and [tex]\((-6, -1)\)[/tex]:
[tex]\[
m = \frac{-1 - 1}{-6 - (-1)} = \frac{-1 - 1}{-6 + 1} = \frac{-2}{-5} = 0.4
\][/tex]
This slope is [tex]\(0.4\)[/tex], so this pair is parallel.
- For [tex]\((0, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]:
[tex]\[
m = \frac{5 - 0}{2 - 0} = \frac{5}{2} = 2.5
\][/tex]
This slope is [tex]\(2.5\)[/tex], not [tex]\(0.4\)[/tex], so this pair is not parallel.
- For [tex]\((1, 0)\)[/tex] and [tex]\((6, 2)\)[/tex]:
[tex]\[
m = \frac{2 - 0}{6 - 1} = \frac{2}{5} = 0.4
\][/tex]
This slope is [tex]\(0.4\)[/tex], so this pair is parallel.
- For [tex]\((3, 0)\)[/tex] and [tex]\((8, 2)\)[/tex]:
[tex]\[
m = \frac{2 - 0}{8 - 3} = \frac{2}{5} = 0.4
\][/tex]
This slope is [tex]\(0.4\)[/tex], so this pair is parallel.
Therefore, the pairs that could be points on a line parallel to the line containing [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] are:
[tex]\[
(-1, 1) \text{ and } (-6, -1),\ (1, 0) \text{ and } (6, 2),\ (3, 0) \text{ and } (8, 2).
\][/tex]
