Answer :
To determine which ordered pairs could be points on a line parallel to a given line with a slope of [tex]\(-\frac{3}{5}\)[/tex], we need to identify pairs of points that yield the same slope when a line is drawn through them.
For each pair of points, we will calculate the slope using the formula for the slope 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]
Let's analyze each pair:
1. For the points [tex]\((-8, 8)\)[/tex] and [tex]\((2, 2)\)[/tex]:
[tex]\[ m = \frac{2 - 8}{2 - (-8)} = \frac{-6}{10} = -\frac{3}{5} \][/tex]
This pair has a slope of [tex]\(-\frac{3}{5}\)[/tex], which matches the given slope.
2. For the points [tex]\((-5, -1)\)[/tex] and [tex]\((0, 2)\)[/tex]:
[tex]\[ m = \frac{2 - (-1)}{0 - (-5)} = \frac{3}{5} \][/tex]
This pair has a slope of [tex]\(\frac{3}{5}\)[/tex], which does not match the given slope.
3. For the points [tex]\((-3, 6)\)[/tex] and [tex]\((6, -9)\)[/tex]:
[tex]\[ m = \frac{-9 - 6}{6 - (-3)} = \frac{-15}{9} = -\frac{5}{3} \][/tex]
This pair has a slope of [tex]\(-\frac{5}{3}\)[/tex], which does not match the given slope.
4. For the points [tex]\((-2, 1)\)[/tex] and [tex]\((3, -2)\)[/tex]:
[tex]\[ m = \frac{-2 - 1}{3 - (-2)} = \frac{-3}{5} = -\frac{3}{5} \][/tex]
This pair has a slope of [tex]\(-\frac{3}{5}\)[/tex], which matches the given slope.
5. For the points [tex]\((0, 2)\)[/tex] and [tex]\((5, 5)\)[/tex]:
[tex]\[ m = \frac{5 - 2}{5 - 0} = \frac{3}{5} \][/tex]
This pair has a slope of [tex]\(\frac{3}{5}\)[/tex], which does not match the given slope.
After analyzing each pair, the pairs whose slopes match the given slope of [tex]\(-\frac{3}{5}\)[/tex] are:
1. [tex]\((-8, 8)\)[/tex] and [tex]\((2, 2)\)[/tex]
2. [tex]\((-2, 1)\)[/tex] and [tex]\((3, -2)\)[/tex]
Therefore, the two options that could be points on a parallel line are:
[tex]\((-8, 8)\)[/tex] and [tex]\((2, 2)\)[/tex], as well as [tex]\((-2, 1)\)[/tex] and [tex]\((3, -2)\)[/tex].
For each pair of points, we will calculate the slope using the formula for the slope 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]
Let's analyze each pair:
1. For the points [tex]\((-8, 8)\)[/tex] and [tex]\((2, 2)\)[/tex]:
[tex]\[ m = \frac{2 - 8}{2 - (-8)} = \frac{-6}{10} = -\frac{3}{5} \][/tex]
This pair has a slope of [tex]\(-\frac{3}{5}\)[/tex], which matches the given slope.
2. For the points [tex]\((-5, -1)\)[/tex] and [tex]\((0, 2)\)[/tex]:
[tex]\[ m = \frac{2 - (-1)}{0 - (-5)} = \frac{3}{5} \][/tex]
This pair has a slope of [tex]\(\frac{3}{5}\)[/tex], which does not match the given slope.
3. For the points [tex]\((-3, 6)\)[/tex] and [tex]\((6, -9)\)[/tex]:
[tex]\[ m = \frac{-9 - 6}{6 - (-3)} = \frac{-15}{9} = -\frac{5}{3} \][/tex]
This pair has a slope of [tex]\(-\frac{5}{3}\)[/tex], which does not match the given slope.
4. For the points [tex]\((-2, 1)\)[/tex] and [tex]\((3, -2)\)[/tex]:
[tex]\[ m = \frac{-2 - 1}{3 - (-2)} = \frac{-3}{5} = -\frac{3}{5} \][/tex]
This pair has a slope of [tex]\(-\frac{3}{5}\)[/tex], which matches the given slope.
5. For the points [tex]\((0, 2)\)[/tex] and [tex]\((5, 5)\)[/tex]:
[tex]\[ m = \frac{5 - 2}{5 - 0} = \frac{3}{5} \][/tex]
This pair has a slope of [tex]\(\frac{3}{5}\)[/tex], which does not match the given slope.
After analyzing each pair, the pairs whose slopes match the given slope of [tex]\(-\frac{3}{5}\)[/tex] are:
1. [tex]\((-8, 8)\)[/tex] and [tex]\((2, 2)\)[/tex]
2. [tex]\((-2, 1)\)[/tex] and [tex]\((3, -2)\)[/tex]
Therefore, the two options that could be points on a parallel line are:
[tex]\((-8, 8)\)[/tex] and [tex]\((2, 2)\)[/tex], as well as [tex]\((-2, 1)\)[/tex] and [tex]\((3, -2)\)[/tex].
