Finding The Equation Of A Line Passing Through (2, 11) And (-8, -19)

In the realm of coordinate geometry, determining the equation of a line is a fundamental concept. Given two points, there exists a unique straight line that passes through them. This article delves into the process of finding the equation of a line when two points on the line are known. Specifically, we will focus on the points (2, 11) and (-8, -19). Understanding how to derive the equation of a line is crucial for various applications in mathematics, physics, engineering, and computer graphics. The equation of a line provides a concise way to represent the relationship between the x and y coordinates of points lying on the line. There are several forms of linear equations, including slope-intercept form, point-slope form, and standard form. In this article, we will primarily use the slope-intercept form, which is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept is the point where the line intersects the y-axis. Calculating the slope is the first crucial step in determining the equation of the line. The slope is defined as the change in y divided by the change in x between any two points on the line. Once the slope is known, we can use either the slope-intercept form or the point-slope form to find the equation of the line. The point-slope form is particularly useful when we have the slope and a single point on the line. By substituting the coordinates of the point and the slope into the point-slope form, we can derive the equation of the line in slope-intercept form. This article provides a step-by-step guide to finding the equation of the line passing through the points (2, 11) and (-8, -19). We will start by calculating the slope using the given coordinates, then use the slope-intercept form to determine the equation of the line. Additionally, we will discuss the importance of understanding linear equations in various contexts and their practical applications. By the end of this article, you will have a solid understanding of how to find the equation of a line given two points and appreciate the significance of this concept in mathematics and beyond. Furthermore, you will learn how to apply these principles to solve similar problems and gain confidence in your ability to work with linear equations. This knowledge will serve as a foundation for more advanced topics in algebra and calculus, where linear equations are frequently used to model real-world phenomena and solve complex problems. This skill is not only essential for academic pursuits but also for practical applications in various fields, where understanding linear relationships can help in making informed decisions and predictions.

Calculating the Slope

To find the equation of the line, the initial step involves calculating the slope (m). The slope represents the rate of change of y with respect to x and is a crucial parameter in defining the line's direction and steepness. 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 given points are (2, 11) and (-8, -19). Let's assign (x₁, y₁) = (2, 11) and (x₂, y₂) = (-8, -19). Now, we can substitute these values into the slope formula: m = (-19 - 11) / (-8 - 2). Simplifying the numerator, we have -19 - 11 = -30. Simplifying the denominator, we have -8 - 2 = -10. Therefore, the slope m = -30 / -10. Dividing -30 by -10 gives us m = 3. So, the slope of the line passing through the points (2, 11) and (-8, -19) is 3. A positive slope indicates that the line rises as we move from left to right. The magnitude of the slope, in this case, 3, tells us how steep the line is. A slope of 3 means that for every 1 unit increase in x, the y-value increases by 3 units. Understanding the slope is fundamental to grasping the behavior of linear equations. It allows us to predict how the dependent variable (y) changes in response to changes in the independent variable (x). In various real-world applications, the slope can represent different things, such as the rate of change of velocity in physics, the cost per unit in economics, or the growth rate in biology. Once we have calculated the slope, we can use it along with one of the given points to determine the equation of the line. This is typically done using either the point-slope form or the slope-intercept form of the equation. In the next section, we will use the slope we just calculated and one of the points to find the equation of the line in slope-intercept form, which is y = mx + b. This form is particularly useful because it directly provides the slope (m) and the y-intercept (b), allowing us to easily visualize and analyze the line. The process of calculating the slope and using it to find the equation of the line is a fundamental skill in algebra and is applied in various areas of mathematics and science.

Using Slope-Intercept Form to Find the Equation

Now that we have calculated the slope, which is m = 3, the next step is to find the equation of the line using the slope-intercept form, y = mx + b. The slope-intercept form is a convenient way to represent a linear equation because it clearly shows the slope (m) and the y-intercept (b) of the line. The y-intercept is the point where the line intersects the y-axis, and its value is given by b. To find the value of b, we can substitute the slope (m = 3) and the coordinates of one of the given points into the equation y = mx + b. Let's use the point (2, 11). Substituting x = 2 and y = 11 into the equation, we get: 11 = 3(2) + b. Simplifying, we have 11 = 6 + b. To isolate b, we subtract 6 from both sides of the equation: 11 - 6 = b. This gives us b = 5. So, the y-intercept of the line is 5. Now that we have both the slope (m = 3) and the y-intercept (b = 5), we can write the equation of the line in slope-intercept form: y = 3x + 5. This equation represents the line that passes through the points (2, 11) and (-8, -19). To verify that this equation is correct, we can substitute the coordinates of the other point, (-8, -19), into the equation and see if it holds true. Substituting x = -8 into the equation y = 3x + 5, we get: y = 3(-8) + 5. Simplifying, we have y = -24 + 5, which gives us y = -19. Since the calculated y-value matches the y-coordinate of the point (-8, -19), we can be confident that the equation y = 3x + 5 is indeed the equation of the line passing through both points. The slope-intercept form is a powerful tool for analyzing linear relationships. It allows us to quickly identify the slope and y-intercept of a line, which can be useful in various applications. For example, in economics, the slope might represent the marginal cost of production, while the y-intercept might represent the fixed costs. Understanding the slope and y-intercept can provide valuable insights into the behavior of the system being modeled. In addition to the slope-intercept form, there are other forms of linear equations, such as the point-slope form and the standard form. Each form has its advantages and is useful in different situations. However, the slope-intercept form is particularly intuitive and is widely used for its simplicity and clarity. By mastering the process of finding the equation of a line in slope-intercept form, you gain a fundamental skill that is applicable in various areas of mathematics, science, and engineering.

Final Equation and Conclusion

In conclusion, the equation of the line passing through the points (2, 11) and (-8, -19) is y = 3x + 5. This equation was derived by first calculating the slope (m) using the formula m = (y₂ - y₁) / (x₂ - x₁), which yielded a slope of 3. Subsequently, we used the slope-intercept form of a linear equation, y = mx + b, and substituted one of the points, (2, 11), along with the calculated slope, to find the y-intercept (b). By solving the equation 11 = 3(2) + b, we determined that b = 5. Thus, the equation of the line is y = 3x + 5. This result demonstrates the systematic approach to finding the equation of a line given two points. The process involves calculating the slope, using the slope and a point to find the y-intercept, and then expressing the equation in slope-intercept form. The slope-intercept form is particularly useful as it provides a clear understanding of the line's characteristics, with the slope indicating its steepness and direction, and the y-intercept indicating where the line crosses the y-axis. Understanding how to determine the equation of a line is a fundamental skill in algebra and coordinate geometry. It has applications in various fields, including physics, engineering, computer graphics, and economics. For instance, in physics, linear equations can be used to describe the motion of an object at a constant velocity. In engineering, they can be used to model the relationship between different variables in a system. In computer graphics, they are used to draw lines and shapes on the screen. In economics, linear equations can be used to represent supply and demand curves. The ability to find the equation of a line is also crucial for solving more complex problems in mathematics. It is a building block for topics such as systems of equations, linear inequalities, and calculus. Furthermore, it helps in developing problem-solving skills and analytical thinking, which are valuable in any field. The process of finding the equation of a line involves several steps, each of which requires careful attention to detail. It is important to understand the underlying concepts and formulas and to practice applying them to different problems. By mastering this skill, you will be well-equipped to tackle a wide range of mathematical challenges and to apply your knowledge in real-world situations. The equation y = 3x + 5 is a concise and powerful representation of the line passing through the points (2, 11) and (-8, -19). It encapsulates the relationship between the x and y coordinates of all points on the line and provides a foundation for further analysis and applications.

y = 3x + 5