To determine which pairs of points could lie on a line parallel to a line with a slope of [tex]\(-\frac{1}{5}\)[/tex], we need to calculate the slope of the line formed by each pair of points. Lines are parallel if and only if they have the same slope.
First, recall the formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
We will calculate the slope for each pair of points given:
1. For the points [tex]\((-8, 8)\)[/tex] and [tex]\( (2, 2) \)[/tex]:
[tex]\[
m = \frac{2 - 8}{2 - (-8)} = \frac{-6}{10} = -0.6
\][/tex]
2. For the points [tex]\( (5, -1) \)[/tex] and [tex]\( (0, 0) \)[/tex]:
[tex]\[
m = \frac{0 - (-1)}{0 - 5} = \frac{1}{-5} = -0.2
\][/tex]
3. For the points [tex]\((-3, 6)\)[/tex] and [tex]\( (6, -9) \)[/tex]:
[tex]\[
m = \frac{-9 - 6}{6 - (-3)} = \frac{-15}{9} = -1.6667
\][/tex]
4. For the points [tex]\((-2, 1)\)[/tex] and [tex]\((3, -2)\)[/tex]:
[tex]\[
m = \frac{-2 - 1}{3 - (-2)} = \frac{-3}{5} = -0.6
\][/tex]
5. For the points [tex]\((0, 2)\)[/tex] and [tex]\((5, 5)\)[/tex]:
[tex]\[
m = \frac{5 - 2}{5 - 0} = \frac{3}{5} = 0.6
\][/tex]
Now, we compare each calculated slope to [tex]\(-\frac{1}{5}\)[/tex] (which is [tex]\(-0.2\)[/tex] in decimal form). The only pair that has a slope that matches [tex]\(-0.2\)[/tex] is the second pair of points:
Thus, the pairs of points that could lie on a line parallel to the line with slope [tex]\(-\frac{1}{5}\)[/tex] are:
[tex]\[
\boxed{(5, -1) \text{ and } (0, 0)}
\][/tex]
There is only one correct option among the given pairs.
