What Is The Slope Of The Line Through (-7,-8)(7,8)A. 12/7B.7/12C.-12/7D.-7/12
Understanding the concept of slope is fundamental in coordinate geometry, especially when analyzing the characteristics of lines on a graph. If you've ever wondered how to find the slope of a line passing through two specific points—such as (-7, -8) and (7, 8)—this article will guide you through the process step by step. We’ll explore how to calculate the slope, interpret its significance, and examine the multiple-choice options provided: A. 12/7, B. 7/12, C. -12/7, D. -7/12. By the end, you'll have a clear understanding of how to determine the slope of this particular line and apply similar methods to future problems.
What Is the Slope of a Line?
Definition of Slope
The slope of a line measures its steepness and direction. It is commonly represented by the letter "m" in algebra and coordinate geometry. The slope indicates how much the y-coordinate (vertical change) varies with respect to the x-coordinate (horizontal change) as you move along the line.Mathematically, the slope between two points \((x1, y1)\) and \((x2, y2)\) is calculated using the formula:
\[
m = \frac{y2 - y1}{x2 - x1}
\]
This ratio reveals whether the line rises (\(m > 0\)), falls (\(m < 0\)), or is horizontal (\(m = 0\)).
Calculating the Slope Between Two Points
Step-by-Step Process
Let's apply this formula to the specific points given: (-7, -8) and (7, 8).Given:
- Point 1: \((x1, y1) = (-7, -8)\)
- Point 2: \((x2, y2) = (7, 8)\)
Calculate the differences:
- Change in y: \(y2 - y1 = 8 - (-8) = 8 + 8 = 16\)
- Change in x: \(x2 - x1 = 7 - (-7) = 7 + 7 = 14\)
Apply the formula:
\[
m = \frac{16}{14}
\]
Simplify the fraction:
\[
m = \frac{8}{7}
\]
But note, in the options, the fractions are given as 12/7, 7/12, -12/7, -7/12, which suggests a different calculation or a need to double-check our steps.
Re-evaluating the Calculation for Accuracy
Double-Check the Calculations
Let’s verify the differences carefully:- \(y2 - y1 = 8 - (-8) = 8 + 8 = 16\)
- \(x2 - x1 = 7 - (-7) = 7 + 7 = 14\)
This matches our initial calculation, which simplifies to \(\frac{8}{7}\). However, this does not directly match any options provided, so perhaps the options are based on a different interpretation or a different pairing of points.
Important note: The options given are 12/7, 7/12, -12/7, -7/12. Our calculated slope is 8/7, which suggests that either the options are rounded or that the points are not correctly interpreted.
Let's consider the possibility that the points are ordered differently or that the slope is negative.
But since the coordinates are:
- \(x1 = -7\), \(y1 = -8\)
- \(x2 = 7\), \(y2 = 8\)
The differences are consistent with our calculation.
Therefore, the slope is \(\frac{8}{7}\).
Given the options, it appears the closest match is none, but perhaps the options are based on a different calculation.
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Note: Since the options are fractions involving 12/7 and 7/12, perhaps the question is designed to test understanding of the ratio, or perhaps the points are meant to be interpreted differently.
Alternatively, could the options be for the slope of the line through points (-7, -8) and (7, 8), with the options given as fractions?
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Matching Our Calculation to the Given Options
Given that our calculated slope is \(\frac{8}{7}\), let's analyze the options:
- A. 12/7 — close but not exact
- B. 7/12 — reciprocal of 12/7
- C. -12/7 — negative of 12/7
- D. -7/12 — negative reciprocal
Since our slope is \(\frac{8}{7}\), none of these options exactly match. However, perhaps the problem expects us to identify the slope as \(\frac{12}{7}\) or \(\frac{7}{12}\), which suggests a different pairing of points or a different calculation.
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Let's visualize the points to confirm the slope:
- The change in y from (-7, -8) to (7, 8): 16
- The change in x: 14
The ratio is \(16/14 = 8/7\).
Therefore, the slope is \(\frac{8}{7}\).
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Understanding the Multiple Choice Options
Given the options, it appears there might be a misprint or a different interpretation. Let's analyze the options to understand what slopes they represent:
- Option A: 12/7
- Option B: 7/12
- Option C: -12/7
- Option D: -7/12
Notice that options C and D are negatives of options A and B, respectively.
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Conclusion: Determining the Correct Slope
Based on the calculations:
- The slope between the points (-7, -8) and (7, 8) is \(\frac{8}{7}\).
- None of the options directly match \(\frac{8}{7}\).
- The options provided are 12/7, 7/12, -12/7, and -7/12.
- Given the calculations, the closest match is A. 12/7, which suggests that perhaps the problem or options are simplified or rounded.
However, based on precise calculation, the slope is \(\frac{8}{7}\), which is not among the options.
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Final Thoughts and Practical Tips
When solving for the slope between two points:
- Use the formula \(m = \frac{y2 - y1}{x2 - x1}\).
- Carefully subtract the coordinates, paying attention to signs.
- Simplify the resulting fraction to its lowest terms.
- Compare your result with multiple-choice options carefully.
In this specific problem, the accurate slope between (-7, -8) and (7, 8) is \(\frac{8}{7}\), which suggests that the most appropriate option, if any, would be a close approximation.
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Summary
- The slope of a line is a measure of its steepness.
- It is calculated using the coordinate difference formula.
- For the points (-7, -8) and (7, 8), the slope is \(\frac{8}{7}\).
- The provided options include fractions similar to this, but none exactly match.
- Always double-check calculations and interpretations when selecting from multiple-choice options.
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Remember: Precise calculation and careful comparison to options are key to correctly solving slope-related questions.
