Answer :
To determine which ordered pairs could be points on a line parallel to the one containing the points [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex], we need to follow these steps:
1. Calculate the slope of the original line:
The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex]:
[tex]\[ m = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = 0.4 \][/tex]
So, the slope of the original line is [tex]\(0.4\)[/tex].
2. Check each pair of points to see if they have the same slope:
- For the points [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 not [tex]\(0.4\)[/tex], so this pair does not represent a parallel line.
- For the points [tex]\((-1, 1)\)[/tex] and [tex]\((-6, -1)\)[/tex]:
[tex]\[ m = \frac{-1 - 1}{-6 - (-1)} = \frac{-2}{-6 + 1} = \frac{-2}{-5} = 0.4 \][/tex]
This slope is [tex]\(0.4\)[/tex], so this pair represents a parallel line.
- For the points [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 not [tex]\(0.4\)[/tex], so this pair does not represent a parallel line.
- For the points [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 represents a parallel line.
- For the points [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 represents a parallel line.
In conclusion, the ordered pairs that could be points on a line parallel to the one containing [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] are:
- [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]
1. Calculate the slope of the original line:
The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex]:
[tex]\[ m = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = 0.4 \][/tex]
So, the slope of the original line is [tex]\(0.4\)[/tex].
2. Check each pair of points to see if they have the same slope:
- For the points [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 not [tex]\(0.4\)[/tex], so this pair does not represent a parallel line.
- For the points [tex]\((-1, 1)\)[/tex] and [tex]\((-6, -1)\)[/tex]:
[tex]\[ m = \frac{-1 - 1}{-6 - (-1)} = \frac{-2}{-6 + 1} = \frac{-2}{-5} = 0.4 \][/tex]
This slope is [tex]\(0.4\)[/tex], so this pair represents a parallel line.
- For the points [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 not [tex]\(0.4\)[/tex], so this pair does not represent a parallel line.
- For the points [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 represents a parallel line.
- For the points [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 represents a parallel line.
In conclusion, the ordered pairs that could be points on a line parallel to the one containing [tex]\((3, 4)\)[/tex] and [tex]\((-2, 2)\)[/tex] are:
- [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]
