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Sagot :
To determine which ordered pairs could be points on a line parallel to the line containing the points [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex], we first need to calculate the slope of the line that passes through [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex].
1. Calculate the slope of the line through [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex]
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = 0.4 \][/tex]
Next, we will calculate the slopes of the lines formed by each pair of points and compare them to the calculated slope [tex]\(0.4\)[/tex]. If the slopes are equal, the lines are parallel.
2. Calculate the slopes for given pairs and compare:
- For [tex]\((-2,-5)\)[/tex] and [tex]\((-7,-3)\)[/tex]:
[tex]\[ \text{slope} = \frac{-3 - (-5)}{-7 - (-2)} = \frac{-3 + 5}{-7 + 2} = \frac{2}{-5} = -0.4 \][/tex]
The slope is [tex]\(-0.4\)[/tex]. This is not equal to [tex]\(0.4\)[/tex], therefore, they are not parallel.
- For [tex]\((-1,1)\)[/tex] and [tex]\((-6,-1)\)[/tex]:
[tex]\[ \text{slope} = \frac{-1 - 1}{-6 - (-1)} = \frac{-1 - 1}{-6 + 1} = \frac{-2}{-5} = 0.4 \][/tex]
The slope is [tex]\(0.4\)[/tex]. This is equal to [tex]\(0.4\)[/tex], therefore, they are parallel.
- For [tex]\((0,0)\)[/tex] and [tex]\((2,5)\)[/tex]:
[tex]\[ \text{slope} = \frac{5 - 0}{2 - 0} = \frac{5}{2} = 2.5 \][/tex]
The slope is [tex]\(2.5\)[/tex]. This is not equal to [tex]\(0.4\)[/tex], therefore, they are not parallel.
- For [tex]\((1,0)\)[/tex] and [tex]\((6,2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - 0}{6 - 1} = \frac{2}{5} = 0.4 \][/tex]
The slope is [tex]\(0.4\)[/tex]. This is equal to [tex]\(0.4\)[/tex], therefore, they are parallel.
- For [tex]\((3,0)\)[/tex] and [tex]\((8,2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - 0}{8 - 3} = \frac{2}{5} = 0.4 \][/tex]
The slope is [tex]\(0.4\)[/tex]. This is equal to [tex]\(0.4\)[/tex], therefore, they are parallel.
3. Conclusion:
The ordered pairs that could be points on a line parallel to the line 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]
These are the pairs with slopes equal to [tex]\(0.4\)[/tex].
1. Calculate the slope of the line through [tex]\((3,4)\)[/tex] and [tex]\((-2,2)\)[/tex]
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 4}{-2 - 3} = \frac{-2}{-5} = 0.4 \][/tex]
Next, we will calculate the slopes of the lines formed by each pair of points and compare them to the calculated slope [tex]\(0.4\)[/tex]. If the slopes are equal, the lines are parallel.
2. Calculate the slopes for given pairs and compare:
- For [tex]\((-2,-5)\)[/tex] and [tex]\((-7,-3)\)[/tex]:
[tex]\[ \text{slope} = \frac{-3 - (-5)}{-7 - (-2)} = \frac{-3 + 5}{-7 + 2} = \frac{2}{-5} = -0.4 \][/tex]
The slope is [tex]\(-0.4\)[/tex]. This is not equal to [tex]\(0.4\)[/tex], therefore, they are not parallel.
- For [tex]\((-1,1)\)[/tex] and [tex]\((-6,-1)\)[/tex]:
[tex]\[ \text{slope} = \frac{-1 - 1}{-6 - (-1)} = \frac{-1 - 1}{-6 + 1} = \frac{-2}{-5} = 0.4 \][/tex]
The slope is [tex]\(0.4\)[/tex]. This is equal to [tex]\(0.4\)[/tex], therefore, they are parallel.
- For [tex]\((0,0)\)[/tex] and [tex]\((2,5)\)[/tex]:
[tex]\[ \text{slope} = \frac{5 - 0}{2 - 0} = \frac{5}{2} = 2.5 \][/tex]
The slope is [tex]\(2.5\)[/tex]. This is not equal to [tex]\(0.4\)[/tex], therefore, they are not parallel.
- For [tex]\((1,0)\)[/tex] and [tex]\((6,2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - 0}{6 - 1} = \frac{2}{5} = 0.4 \][/tex]
The slope is [tex]\(0.4\)[/tex]. This is equal to [tex]\(0.4\)[/tex], therefore, they are parallel.
- For [tex]\((3,0)\)[/tex] and [tex]\((8,2)\)[/tex]:
[tex]\[ \text{slope} = \frac{2 - 0}{8 - 3} = \frac{2}{5} = 0.4 \][/tex]
The slope is [tex]\(0.4\)[/tex]. This is equal to [tex]\(0.4\)[/tex], therefore, they are parallel.
3. Conclusion:
The ordered pairs that could be points on a line parallel to the line 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]
These are the pairs with slopes equal to [tex]\(0.4\)[/tex].
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