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 first need to find the slope of the given line. Recall that the slope [tex]\(m\)[/tex] of a line through two 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]
Begin by calculating the slope of the line containing the points [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex].
[tex]\[
m = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = \frac{2}{5}
\][/tex]
So, the slope of the given line is [tex]\(\frac{2}{5}\)[/tex].
For other lines to be parallel to this line, they must have the same slope, [tex]\(\frac{2}{5}\)[/tex]. We will calculate the slope for each of the given pairs and see if it matches [tex]\(\frac{2}{5}\)[/tex].
1. Calculating the slope for [tex]$(-2,-5)$[/tex] and [tex]$(-7,-3)$[/tex]:
[tex]\[
m = \frac{-3 - (-5)}{-7 - (-2)} = \frac{-3 + 5}{-7 + 2} = \frac{2}{-5} = -\frac{2}{5}
\][/tex]
This slope is [tex]\(-\frac{2}{5}\)[/tex], which is not equal to [tex]\(\frac{2}{5}\)[/tex], so these points are not on a parallel line.
2. Calculating the slope for [tex]$(-1,1)$[/tex] and [tex]$(-6,-1)$[/tex]:
[tex]\[
m = \frac{-1 - 1}{-6 - (-1)} = \frac{-1 - 1}{-6 + 1} = \frac{-2}{-5} = \frac{2}{5}
\][/tex]
This slope is [tex]\(\frac{2}{5}\)[/tex], which matches the original slope. So these points are on a parallel line.
3. Calculating the slope for [tex]$(0,0)$[/tex] and [tex]$(2,5)$[/tex]:
[tex]\[
m = \frac{5 - 0}{2 - 0} = \frac{5}{2}
\][/tex]
This slope is [tex]\(\frac{5}{2}\)[/tex], which is not equal to [tex]\(\frac{2}{5}\)[/tex], so these points are not on a parallel line.
4. Calculating the slope for [tex]$(1,0)$[/tex] and [tex]$(6,2)$[/tex]:
[tex]\[
m = \frac{2 - 0}{6 - 1} = \frac{2}{5}
\][/tex]
This slope is [tex]\(\frac{2}{5}\)[/tex], which matches the original slope. So these points are on a parallel line.
5. Calculating the slope for [tex]$(3,0)$[/tex] and [tex]$(8,2)$[/tex]:
[tex]\[
m = \frac{2 - 0}{8 - 3} = \frac{2}{5}
\][/tex]
This slope is [tex]\(\frac{2}{5}\)[/tex], which matches the original slope. So these points are on a parallel line.
After checking each pair, we find that the pairs [tex]\((-1,1)\)[/tex] and [tex]\((-6,-1)\)[/tex], [tex]\((1,0)\)[/tex] and [tex]\((6,2)\)[/tex], and [tex]\((3,0)\)[/tex] and [tex]\((8,2)\)[/tex] could be points on a line parallel to the line containing [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex]. Thus, the pairs that satisfy this condition are:
[tex]\[
\boxed{(-1,1) \text{ and } (-6,-1), (1,0) \text{ and } (6,2), (3,0) \text{ and } (8,2)}
\][/tex]
