To determine which sets of ordered pairs have a slope of 2, we need to calculate the slope for each pair and see if it matches our target slope of 2. The formula for the slope [tex]\( m \)[/tex] between 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]
Let's go through each pair step-by-step:
1. Pair [tex]\((1, -4)\)[/tex] and [tex]\((2, 6)\)[/tex]:
- [tex]\[ m = \frac{6 - (-4)}{2 - 1} = \frac{6 + 4}{2 - 1} = \frac{10}{1} = 10 \][/tex]
- The slope is 10, so this pair does not have a slope of 2.
2. Pair [tex]\((3, 4)\)[/tex] and [tex]\((5, 8)\)[/tex]:
- [tex]\[ m = \frac{8 - 4}{5 - 3} = \frac{8 - 4}{2} = \frac{4}{2} = 2 \][/tex]
- The slope is 2, so this pair does have a slope of 2.
3. Pair [tex]\((2, 5)\)[/tex] and [tex]\((2, 7)\)[/tex]:
- [tex]\[ m = \frac{7 - 5}{2 - 2} = \frac{2}{0} \][/tex]
- Division by zero occurs, so the slope is undefined and not equal to 2.
4. Pair [tex]\((5, 7)\)[/tex] and [tex]\((4, 5)\)[/tex]:
- [tex]\[ m = \frac{5 - 7}{4 - 5} = \frac{5 - 7}{4 - 5} = \frac{-2}{-1} = 2 \][/tex]
- The slope is 2, so this pair does have a slope of 2.
5. Pair [tex]\((-2, 4)\)[/tex] and [tex]\((0, 5)\)[/tex]:
- [tex]\[ m = \frac{5 - 4}{0 - (-2)} = \frac{5 - 4}{0 + 2} = \frac{1}{2} \][/tex]
- The slope is 0.5, so this pair does not have a slope of 2.
6. Pair [tex]\((-2, 4)\)[/tex] and [tex]\((1, -2)\)[/tex]:
- [tex]\[ m = \frac{-2 - 4}{1 - (-2)} = \frac{-2 - 4}{1 + 2} = \frac{-6}{3} = -2 \][/tex]
- The slope is -2, so this pair does not have a slope of 2.
Thus, the pairs with a slope of 2 are:
- [tex]\((3, 4)\)[/tex] and [tex]\((5, 8)\)[/tex]
- [tex]\((5, 7)\)[/tex] and [tex]\((4, 5)\)[/tex]
So, the ordered pairs that have a slope of 2 are:
[tex]\[
(3, 4), (5, 8) \quad \text{and} \quad (5, 7), (4, 5)
\][/tex]
