To determine which ordered pairs could be points on a line parallel to the line passing through [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex], we first need to find the slope of the line defined by these two points.
1. Calculate the slope of the line passing through [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex]:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = \frac{2}{5}
\][/tex]
2. A line that is parallel to this line will have the same slope, [tex]\(\frac{2}{5}\)[/tex].
3. Now we need to check each of the given pairs and calculate their slopes to see which ones have the same slope [tex]\(\frac{2}{5}\)[/tex].
Let's check each pair:
### Pair: [tex]\((-2, -5)\)[/tex] and [tex]\((-7, -3)\)[/tex]
[tex]\[
\text{slope} = \frac{-3 - (-5)}{-7 - (-2)} = \frac{-3 + 5}{-7 + 2} = \frac{2}{-5} = -\frac{2}{5}
\][/tex]
The slope is [tex]\(-\frac{2}{5}\)[/tex], which is not equal to [tex]\(\frac{2}{5}\)[/tex].
### Pair: [tex]\((-1, 1)\)[/tex] and [tex]\((-6, -1)\)[/tex]
[tex]\[
\text{slope} = \frac{-1 - 1}{-6 - (-1)} = \frac{-1 -1}{-6 + 1} = \frac{-2}{-5} = \frac{2}{5}
\][/tex]
The slope is [tex]\(\frac{2}{5}\)[/tex], which matches the required slope.
### Pair: [tex]\((0, 0)\)[/tex] and [tex]\((2, 5)\)[/tex]
[tex]\[
\text{slope} = \frac{5 - 0}{2 - 0} = \frac{5}{2}
\][/tex]
The slope is [tex]\(\frac{5}{2}\)[/tex], which is not equal to [tex]\(\frac{2}{5}\)[/tex].
### Pair: [tex]\((1, 0)\)[/tex] and [tex]\((6, 2)\)[/tex]
[tex]\[
\text{slope} = \frac{2 - 0}{6 - 1} = \frac{2}{5}
\][/tex]
The slope is [tex]\(\frac{2}{5}\)[/tex], which matches the required slope.
### Pair: [tex]\((3, 0)\)[/tex] and [tex]\((8, 2)\)[/tex]
[tex]\[
\text{slope} = \frac{2 - 0}{8 - 3} = \frac{2}{5}
\][/tex]
The slope is [tex]\(\frac{2}{5}\)[/tex], which matches the required slope.
From these calculations, we determined that the following pairs of points have the same slope [tex]\(\frac{2}{5}\)[/tex] and are therefore on lines parallel to the line passing through [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex]:
- [tex]\((-1, 1)\)[/tex] and [tex]\((-6, -1)\)[/tex]
- [tex]\((1, 0)\)[/tex] and [tex]\((6, 2)\)[/tex]
- [tex]\((3, 0)\)[/tex] and [tex]\((8, 2)\)[/tex]
So, the ordered pairs that could be points on a line parallel to the line that contains [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] are:
[tex]\[
\begin{array}{l}
(-1, 1) \text{ and } (-6, -1) \\
(1, 0) \text{ and } (6, 2) \\
(3, 0) \text{ and } (8, 2)
\end{array}
\][/tex]
