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To determine which set of ordered pairs has the same slope as the given set [tex]$(-1, 5)$[/tex] and [tex]$(2, 8)$[/tex], we need to first calculate the slope of the given set and then compare it with the slopes of the provided options.
Step 1: Calculate the slope of the given set [tex]$(-1, 5)$[/tex] and [tex]$(2, 8)$[/tex]
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:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((-1, 5)\)[/tex] and [tex]\((2, 8)\)[/tex]:
[tex]\[ x_1 = -1, \quad y_1 = 5, \quad x_2 = 2, \quad y_2 = 8 \][/tex]
[tex]\[ m_{\text{given}} = \frac{8 - 5}{2 - (-1)} = \frac{3}{3} = 1 \][/tex]
Step 2: Calculate the slopes of the provided options and compare them
- Option 1: [tex]$(7, -7)$[/tex] and [tex]$(2, 10)$[/tex]
[tex]\[ x_1 = 7, \quad y_1 = -7, \quad x_2 = 2, \quad y_2 = 10 \][/tex]
[tex]\[ m_1 = \frac{10 - (-7)}{2 - 7} = \frac{10 + 7}{2 - 7} = \frac{17}{-5} = -3.4 \][/tex]
- Option 2: [tex]$(-3, 5)$[/tex] and [tex]$(2, 5)$[/tex]
[tex]\[ x_1 = -3, \quad y_1 = 5, \quad x_2 = 2, \quad y_2 = 5 \][/tex]
[tex]\[ m_2 = \frac{5 - 5}{2 - (-3)} = \frac{0}{2 + 3} = \frac{0}{5} = 0 \][/tex]
- Option 3: [tex]$(8, 10)$[/tex] and [tex]$(7, 9)$[/tex]
[tex]\[ x_1 = 8, \quad y_1 = 10, \quad x_2 = 7, \quad y_2 = 9 \][/tex]
[tex]\[ m_3 = \frac{9 - 10}{7 - 8} = \frac{-1}{-1} = 1 \][/tex]
- Option 4: [tex]$(2, 0)$[/tex] and [tex]$(8, 10)$[/tex]
[tex]\[ x_1 = 2, \quad y_1 = 0, \quad x_2 = 8, \quad y_2 = 10 \][/tex]
[tex]\[ m_4 = \frac{10 - 0}{8 - 2} = \frac{10}{6} = \frac{5}{3} \][/tex]
Conclusion: Compare the slopes
We need to identify the option that has the same slope as the given slope [tex]\(m_{\text{given}} = 1\)[/tex]:
- Option 1: [tex]\(m_1 = -3.4\)[/tex]
- Option 2: [tex]\(m_2 = 0\)[/tex]
- Option 3: [tex]\(m_3 = 1\)[/tex] (matching slope)
- Option 4: [tex]\(m_4 = \frac{5}{3}\)[/tex]
The set of ordered pairs that has the same slope as the given set [tex]\((-1, 5)\)[/tex] and [tex]\((2, 8)\)[/tex] is:
[tex]\[ \boxed{(8, 10) \text{ and } (7, 9)} \][/tex]
Therefore, Option 3 is the correct answer. The ordered pairs [tex]\((8, 10)\)[/tex] and [tex]\((7, 9)\)[/tex] have the same slope as the given set [tex]\((-1, 5)\)[/tex] and [tex]\((2, 8)\)[/tex].
Step 1: Calculate the slope of the given set [tex]$(-1, 5)$[/tex] and [tex]$(2, 8)$[/tex]
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:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points [tex]\((-1, 5)\)[/tex] and [tex]\((2, 8)\)[/tex]:
[tex]\[ x_1 = -1, \quad y_1 = 5, \quad x_2 = 2, \quad y_2 = 8 \][/tex]
[tex]\[ m_{\text{given}} = \frac{8 - 5}{2 - (-1)} = \frac{3}{3} = 1 \][/tex]
Step 2: Calculate the slopes of the provided options and compare them
- Option 1: [tex]$(7, -7)$[/tex] and [tex]$(2, 10)$[/tex]
[tex]\[ x_1 = 7, \quad y_1 = -7, \quad x_2 = 2, \quad y_2 = 10 \][/tex]
[tex]\[ m_1 = \frac{10 - (-7)}{2 - 7} = \frac{10 + 7}{2 - 7} = \frac{17}{-5} = -3.4 \][/tex]
- Option 2: [tex]$(-3, 5)$[/tex] and [tex]$(2, 5)$[/tex]
[tex]\[ x_1 = -3, \quad y_1 = 5, \quad x_2 = 2, \quad y_2 = 5 \][/tex]
[tex]\[ m_2 = \frac{5 - 5}{2 - (-3)} = \frac{0}{2 + 3} = \frac{0}{5} = 0 \][/tex]
- Option 3: [tex]$(8, 10)$[/tex] and [tex]$(7, 9)$[/tex]
[tex]\[ x_1 = 8, \quad y_1 = 10, \quad x_2 = 7, \quad y_2 = 9 \][/tex]
[tex]\[ m_3 = \frac{9 - 10}{7 - 8} = \frac{-1}{-1} = 1 \][/tex]
- Option 4: [tex]$(2, 0)$[/tex] and [tex]$(8, 10)$[/tex]
[tex]\[ x_1 = 2, \quad y_1 = 0, \quad x_2 = 8, \quad y_2 = 10 \][/tex]
[tex]\[ m_4 = \frac{10 - 0}{8 - 2} = \frac{10}{6} = \frac{5}{3} \][/tex]
Conclusion: Compare the slopes
We need to identify the option that has the same slope as the given slope [tex]\(m_{\text{given}} = 1\)[/tex]:
- Option 1: [tex]\(m_1 = -3.4\)[/tex]
- Option 2: [tex]\(m_2 = 0\)[/tex]
- Option 3: [tex]\(m_3 = 1\)[/tex] (matching slope)
- Option 4: [tex]\(m_4 = \frac{5}{3}\)[/tex]
The set of ordered pairs that has the same slope as the given set [tex]\((-1, 5)\)[/tex] and [tex]\((2, 8)\)[/tex] is:
[tex]\[ \boxed{(8, 10) \text{ and } (7, 9)} \][/tex]
Therefore, Option 3 is the correct answer. The ordered pairs [tex]\((8, 10)\)[/tex] and [tex]\((7, 9)\)[/tex] have the same slope as the given set [tex]\((-1, 5)\)[/tex] and [tex]\((2, 8)\)[/tex].
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