Equation Of Line GH Through Points (2, 5) And (6, 9)

Jul 9, 2025 by ADMIN 53 views

Finding the Equation of a Line Passing Through Two Points

In mathematics, determining the equation of a line is a fundamental concept in coordinate geometry. Given two points on a line, we can find the equation of the line in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. This article will delve into the step-by-step process of finding the equation of a line, using the specific example of line GH passing through the points (2, 5) and (6, 9). We will explore the concepts of slope, y-intercept, and how to apply these to formulate the equation of the line. This is an essential skill in algebra and has practical applications in various fields such as physics, engineering, and economics. Understanding how to calculate the equation of a line from two points is crucial for solving problems related to linear relationships and modeling real-world scenarios. The process involves first calculating the slope, which represents the rate of change of y with respect to x. Then, using one of the given points and the calculated slope, we can determine the y-intercept, which is the point where the line crosses the y-axis. Finally, with both the slope and y-intercept, we can write the equation of the line in slope-intercept form. This article will guide you through each step with clear explanations and examples, making it easy to grasp the concepts and apply them to similar problems. By the end of this discussion, you will have a solid understanding of how to find the equation of a line given two points, a skill that is invaluable in mathematics and beyond. This knowledge not only helps in solving mathematical problems but also enhances your analytical and problem-solving abilities, which are essential in various aspects of life. This comprehensive guide ensures that you understand the underlying principles and can confidently tackle any similar question. The ability to find the equation of a line is a building block for more advanced mathematical concepts and is therefore a crucial skill to master.

Calculating the Slope

To begin, we need to calculate the slope (m) of line GH. The slope represents the steepness of the line and is defined as the change in y divided by the change in x. Given two points (x₁, y₁) and (x₂, y₂), the slope can be calculated using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

In our case, the points are (2, 5) and (6, 9). Let's assign (2, 5) as (x₁, y₁) and (6, 9) as (x₂, y₂). Plugging these values into the formula, we get:

m = (9 - 5) / (6 - 2) m = 4 / 4 m = 1

Therefore, the slope of line GH is 1. The slope is a crucial element in determining the equation of a line, as it tells us how much the line rises or falls for each unit increase in the x-coordinate. A positive slope, as we have here, indicates that the line is increasing, meaning that as x increases, y also increases. The magnitude of the slope tells us how steep the line is; a larger slope means a steeper line. In this case, a slope of 1 means that for every unit increase in x, y increases by 1 unit. This information is vital for visualizing the line and understanding its behavior. Knowing the slope is the first step in finding the equation of the line, and it lays the groundwork for the next step, which is determining the y-intercept. Understanding the concept of slope is fundamental not only in mathematics but also in various real-world applications, such as calculating the rate of change in physical quantities or analyzing trends in data. Therefore, mastering this concept is essential for a strong foundation in mathematical problem-solving. The slope provides valuable information about the direction and steepness of the line, making it a cornerstone of linear equations.

Determining the Y-Intercept

Now that we have the slope (m = 1), we need to find the y-intercept (b). The y-intercept is the point where the line intersects the y-axis, which occurs when x = 0. To find the y-intercept, we can use the slope-intercept form of the equation of a line, which is:

y = mx + b

We can plug in the slope (m = 1) and one of the given points, for example, (2, 5), into this equation to solve for b:

5 = (1)(2) + b 5 = 2 + b b = 5 - 2 b = 3

Thus, the y-intercept of line GH is 3. The y-intercept is another key component of the equation of a line. It tells us where the line crosses the y-axis, which is a crucial piece of information for graphing the line and understanding its position in the coordinate plane. In this case, the y-intercept of 3 means that the line GH passes through the point (0, 3) on the y-axis. This value is essential for writing the complete equation of the line. The process of finding the y-intercept involves using the slope we calculated earlier and one of the given points. This method is reliable and straightforward, ensuring that we can accurately determine the y-intercept. The y-intercept, along with the slope, completely defines the line, allowing us to predict its behavior and position for any given value of x. Understanding the concept of the y-intercept is important not only in mathematics but also in various real-world contexts. For example, in a linear model of cost, the y-intercept might represent the fixed costs, while the slope represents the variable costs. Therefore, knowing how to find the y-intercept is a valuable skill with practical applications. The y-intercept, combined with the slope, provides a complete picture of the line, making it possible to write its equation and analyze its properties.

Writing the Equation of the Line

With the slope (m = 1) and the y-intercept (b = 3) determined, we can now write the equation of line GH in slope-intercept form:

y = mx + b

Substituting the values we found, we get:

y = (1)x + 3 y = x + 3

Therefore, the equation of line GH is y = x + 3. This equation represents the relationship between x and y for all points on line GH. It tells us that for any value of x, we can find the corresponding value of y by adding 3 to x. This equation is a concise and powerful way to describe the line and its behavior. The slope-intercept form is particularly useful because it directly shows the slope and y-intercept, making it easy to visualize the line and its characteristics. In this case, we can see that the line has a slope of 1, meaning it rises one unit for every unit increase in x, and a y-intercept of 3, meaning it crosses the y-axis at the point (0, 3). This equation can be used to solve various problems related to line GH, such as finding points on the line, determining whether a given point lies on the line, or comparing it with other lines. The process of writing the equation of a line involves combining the slope and y-intercept into the slope-intercept form. This is a fundamental skill in algebra and is essential for solving a wide range of problems. The equation of the line provides a complete description of the line's properties and behavior, making it a valuable tool for mathematical analysis and problem-solving. Mastering this skill is crucial for a strong foundation in mathematics and its applications.

Conclusion

In conclusion, we have successfully found the equation of line GH passing through the points (2, 5) and (6, 9). The process involved calculating the slope using the formula m = (y₂ - y₁) / (x₂ - x₁), determining the y-intercept by plugging the slope and a point into the equation y = mx + b, and finally, writing the equation of the line in slope-intercept form. We found that the equation of line GH is y = x + 3. This exercise demonstrates the fundamental principles of coordinate geometry and the importance of understanding slope and y-intercept. The ability to find the equation of a line given two points is a crucial skill in mathematics and has numerous applications in various fields. This process not only helps in solving mathematical problems but also enhances analytical and problem-solving abilities. Understanding the concepts of slope and y-intercept allows us to describe and analyze linear relationships effectively. The slope tells us about the steepness and direction of the line, while the y-intercept tells us where the line crosses the y-axis. Together, these two values completely define the line and allow us to write its equation. This equation can then be used to predict the behavior of the line and solve related problems. Mastering this skill is essential for a strong foundation in mathematics and its applications. The process of finding the equation of a line is a building block for more advanced mathematical concepts and is therefore a crucial skill to master. By following the steps outlined in this article, you can confidently tackle similar problems and deepen your understanding of linear equations. The knowledge gained from this discussion will be invaluable in your mathematical journey and beyond.