Finding The Slope Of A Line Passing Through Two Points (-3, -6) And (4, 7)

In mathematics, determining the slope of a line is a fundamental concept in coordinate geometry. The slope provides crucial information about the line's direction and steepness. Specifically, the slope, often denoted by m, quantifies the change in the vertical direction (rise) for every unit change in the horizontal direction (run). In simpler terms, it tells us how much the line goes up or down for every step we take to the right. This article will guide you through the process of calculating the slope of a line given two points, using a clear and step-by-step approach. We will explore the formula, discuss its application with an example, and interpret the result in the context of the line's properties. Understanding the concept of slope is crucial not only in mathematics but also in various real-world applications, such as physics, engineering, and economics, where linear relationships are prevalent. For instance, in physics, the slope of a distance-time graph represents the velocity of an object, while in economics, it can represent the marginal cost or revenue. Therefore, mastering the calculation and interpretation of slope is a valuable skill.

Understanding the Slope Formula

The slope of a line passing through two points, often denoted as (x₁, y₁) and (x₂, y₂), can be calculated using a specific formula. This formula is derived from the fundamental concept of rise over run, where rise represents the vertical change (change in y-coordinates) and run represents the horizontal change (change in x-coordinates). The formula provides a straightforward method for quantifying the steepness and direction of a line on a coordinate plane. By understanding and applying this formula, we can easily determine the slope of any line, given the coordinates of two points on that line. This mathematical tool is essential for analyzing linear relationships and making predictions based on linear models. For example, in statistics, the slope of a regression line indicates the average change in the dependent variable for each unit increase in the independent variable. In finance, the slope of a trend line can help predict future stock prices or economic growth rates. Thus, the slope formula is a powerful tool with widespread applications.

The slope formula, a cornerstone of coordinate geometry, is expressed as follows:

m = (y₂ - y₁) / (x₂ - x₁)

Where:

• m represents the slope of the line.
• (x₁, y₁) and (x₂, y₂) are the coordinates of two distinct points on the line.

This formula is derived from the concept of rise over run, which is a visual way to understand slope. The rise (y₂ - y₁) represents the vertical change between the two points, while the run (x₂ - x₁) represents the horizontal change. The ratio of these changes gives us the slope, which indicates how steeply the line is inclined. A positive slope indicates that the line rises as it moves from left to right, while a negative slope indicates that the line falls. A slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line. The slope formula is not just a mathematical abstraction; it has practical applications in various fields. For instance, architects use slope calculations to design ramps and roofs, ensuring they meet safety and functionality requirements. Engineers use slope to analyze the stability of structures and the flow of fluids in pipes. Even everyday tasks like driving involve an understanding of slope, as drivers must adjust their speed and steering based on the incline of the road.

Applying the Slope Formula to the Given Points

Now, let's apply this slope formula to the specific points given in the problem: (-3, -6) and (4, 7). We will substitute these coordinates into the formula to calculate the slope of the line that passes through these two points. It's crucial to correctly identify which point is (x₁, y₁) and which is (x₂, y₂) to avoid errors in the calculation. The order in which we subtract the coordinates is important; we must subtract the y-coordinates and the x-coordinates in the same order. For example, if we subtract y₁ from y₂, we must also subtract x₁ from x₂. This consistency ensures that we obtain the correct sign for the slope, which indicates the direction of the line. A common mistake is to mix up the order of subtraction, which can lead to an incorrect slope value. Once we have calculated the slope, we can interpret its meaning in terms of the line's steepness and direction. A larger absolute value of the slope indicates a steeper line, while a smaller absolute value indicates a gentler slope. Understanding how to apply the slope formula correctly is essential for solving problems in coordinate geometry and for understanding linear relationships in various contexts. This skill is fundamental for further studies in mathematics and related fields.

1. Identify the coordinates:

   • Let (x₁, y₁) = (-3, -6)
   • Let (x₂, y₂) = (4, 7)

2. Substitute the values into the slope formula:

   • m = (7 - (-6)) / (4 - (-3))

3. Simplify the expression:

   • m = (7 + 6) / (4 + 3)
   • m = 13 / 7

Therefore, the slope of the line containing the points (-3, -6) and (4, 7) is 13/7. This slope calculation demonstrates the practical application of the slope formula. By substituting the coordinates of the two points into the formula and simplifying the expression, we have successfully determined the slope of the line. The result, 13/7, is a positive value, indicating that the line rises as it moves from left to right. The magnitude of the slope also tells us how steep the line is; in this case, a slope of 13/7 suggests a relatively steep line. Understanding this calculation is not just about getting the right answer; it's about comprehending the underlying mathematical concepts. The slope is a fundamental property of a line, and its calculation is essential for various applications, such as graphing linear equations, analyzing linear relationships, and solving problems in physics and engineering. By mastering the slope formula, you are building a solid foundation for more advanced mathematical concepts.

Interpreting the Slope

The calculated slope of 13/7 has a significant interpretation in the context of the line. This positive slope signifies that the line is increasing, meaning it rises as you move from left to right on the coordinate plane. For every 7 units you move horizontally, the line rises 13 units vertically. The magnitude of the slope (13/7) indicates the steepness of the line. A larger slope value implies a steeper line, while a smaller slope value suggests a gentler incline. In this case, 13/7 is greater than 1, indicating a moderately steep line. Visualizing the line on a graph can further enhance understanding. Imagine plotting the two points (-3, -6) and (4, 7) on a coordinate plane and drawing a line through them. The line will clearly ascend from left to right, confirming the positive slope. Furthermore, the steepness of the line can be visually assessed by comparing it to a line with a slope of 1, which would rise 1 unit for every 1 unit of horizontal movement. The line with a slope of 13/7 will be noticeably steeper. Interpreting the slope is not just a mathematical exercise; it provides valuable insights into the relationship between the variables represented on the x and y axes. In real-world applications, the slope can represent various quantities, such as the rate of change of a process, the cost per unit of production, or the velocity of an object. Understanding the meaning of the slope allows us to make predictions and draw conclusions based on the linear relationship between the variables.

The slope, being a ratio, provides valuable insights into the line's behavior. A positive slope, like the one we calculated (13/7), indicates a direct relationship between the x and y variables. In simpler terms, as the x-value increases, the y-value also increases. This can be visualized as an upward-sloping line on a graph. Conversely, a negative slope would indicate an inverse relationship, where an increase in x leads to a decrease in y, resulting in a downward-sloping line. A slope of zero represents a horizontal line, indicating no change in the y-value as x changes. An undefined slope, on the other hand, represents a vertical line, where the x-value remains constant regardless of the y-value. The magnitude of the slope also plays a crucial role in its interpretation. A larger absolute value of the slope signifies a steeper line, indicating a more significant change in y for a given change in x. Conversely, a smaller absolute value indicates a gentler slope, representing a less pronounced change in y for the same change in x. Understanding these nuances of slope interpretation is essential for applying the concept in various fields. For example, in economics, the slope of a supply curve indicates how much the quantity supplied changes in response to a change in price. In physics, the slope of a velocity-time graph represents the acceleration of an object. By mastering the interpretation of slope, you can gain a deeper understanding of the relationships between variables and make informed decisions based on linear models.

Conclusion

In conclusion, we have successfully calculated the slope of the line passing through the points (-3, -6) and (4, 7) using the slope formula. The calculated slope of 13/7 signifies a line that rises steeply from left to right. This exercise demonstrates the importance of understanding and applying fundamental concepts in coordinate geometry. The slope is a crucial property of a line, providing valuable information about its direction and steepness. Mastering the slope formula and its interpretation is essential for various mathematical and real-world applications. From graphing linear equations to analyzing linear relationships in various fields, the concept of slope is a cornerstone of mathematical understanding. This article has provided a step-by-step guide to calculating and interpreting the slope, equipping you with the knowledge and skills to tackle similar problems. By understanding the slope formula and its implications, you can confidently analyze linear relationships and make informed decisions based on graphical representations. The ability to calculate and interpret slope is not just a mathematical skill; it's a valuable tool for problem-solving and critical thinking in various contexts. As you continue your mathematical journey, remember that the slope is a fundamental concept that will continue to be relevant in more advanced topics.

Therefore, the correct answer is:

B. 13/7