Finding The Equation Of A Line Through Points (-4, 11) And (5, -41/2)
Finding the equation of a line that passes through two given points is a fundamental concept in coordinate geometry. This article will delve into the step-by-step process of determining the equation of a line when two points on the line are known. We will use the points (-4, 11) and (5, -41/2) as an example to illustrate the method, ensuring a comprehensive understanding for readers of all backgrounds. The ability to find the equation of a line is crucial in various mathematical and real-world applications, making this a valuable skill to master.
Understanding the Fundamentals
Before we dive into the specifics, let's recap the basic forms of a linear equation. The most common form is the slope-intercept form: y = mx + b, where m represents the slope of the line and b represents the y-intercept. Another useful form is the point-slope form: y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. The point-slope form is particularly helpful when we have the slope and a point, which is precisely the situation we'll encounter when given two points. Understanding these forms is crucial, as they provide the framework for solving our problem. We will use these forms interchangeably to derive the equation of the line passing through the given points. A solid grasp of these fundamentals will allow us to tackle more complex problems in linear algebra and coordinate geometry.
Step 1: Calculating the Slope
The slope of a line, often denoted by m, measures its steepness and direction. It's defined as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. Given two points (x1, y1) and (x2, y2), the slope m can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
In our case, the points are (-4, 11) and (5, -41/2). Let's identify our coordinates:
- x1 = -4
- y1 = 11
- x2 = 5
- y2 = -41/2
Now, we can plug these values into the slope formula:
m = (-41/2 - 11) / (5 - (-4)) m = (-41/2 - 22/2) / (5 + 4) m = (-63/2) / 9 m = -63/18 m = -7/2
Therefore, the slope of the line passing through the points (-4, 11) and (5, -41/2) is -7/2. This negative slope indicates that the line is decreasing as we move from left to right on the coordinate plane. Calculating the slope accurately is a critical first step, as it forms the foundation for finding the equation of the line. A clear understanding of slope allows us to visualize the line's direction and steepness, aiding in our comprehension of linear relationships.
Step 2: Using the Point-Slope Form
Now that we have the slope, m = -7/2, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by:
y - y1 = m(x - x1)
We can use either of the given points, (-4, 11) or (5, -41/2), as (x1, y1). Let's use the point (-4, 11) for simplicity:
- x1 = -4
- y1 = 11
- m = -7/2
Plugging these values into the point-slope form, we get:
y - 11 = (-7/2)(x - (-4)) y - 11 = (-7/2)(x + 4)
This equation represents the line in point-slope form. We can leave it in this form, or we can simplify it further to the slope-intercept form or the standard form. The point-slope form is particularly useful as it directly incorporates the slope and a point on the line, making it a straightforward method for finding the equation. This step highlights the practical application of the point-slope form in solving linear equation problems.
Step 3: Converting to Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form (y = mx + b), we need to distribute the slope and isolate y. Starting from the equation we derived in the previous step:
y - 11 = (-7/2)(x + 4)
First, distribute the -7/2 on the right side:
y - 11 = (-7/2)x + (-7/2)(4) y - 11 = (-7/2)x - 14
Next, add 11 to both sides of the equation to isolate y:
y = (-7/2)x - 14 + 11 y = (-7/2)x - 3
Now, the equation is in slope-intercept form, where m = -7/2 is the slope and b = -3 is the y-intercept. This form is particularly useful for quickly identifying the slope and y-intercept of the line. The conversion to slope-intercept form allows for easy visualization of the line on a graph and facilitates further analysis of its properties.
Step 4: Converting to Standard Form (Optional)
Another common form for linear equations is the standard form, which is written as Ax + By = C, where A, B, and C are integers, and A is non-negative. To convert the equation from slope-intercept form to standard form, we need to eliminate the fraction and rearrange the terms.
Starting from the slope-intercept form:
y = (-7/2)x - 3
First, multiply both sides of the equation by 2 to eliminate the fraction:
2y = 2((-7/2)x - 3) 2y = -7x - 6
Next, add 7x to both sides to move the x term to the left side:
7x + 2y = -6
Now, the equation is in standard form. This form is useful for certain algebraic manipulations and for comparing the equations of different lines. While converting to standard form is optional, it demonstrates a complete understanding of linear equations and their various representations. The standard form provides a concise way to express the relationship between x and y, making it valuable in different mathematical contexts.
Final Answer
The equation of the line passing through the points (-4, 11) and (5, -41/2) can be represented in three forms:
- Point-Slope Form: y - 11 = (-7/2)(x + 4)
- Slope-Intercept Form: y = (-7/2)x - 3
- Standard Form: 7x + 2y = -6
Each of these forms provides a different perspective on the line, but they all represent the same linear relationship. Understanding how to convert between these forms is a valuable skill in mathematics. This comprehensive solution demonstrates the step-by-step process of finding the equation of a line given two points, reinforcing the fundamental concepts of slope, point-slope form, slope-intercept form, and standard form. Mastering these concepts is essential for success in algebra and beyond.
Conclusion
In conclusion, finding the equation of a line passing through two points involves calculating the slope, using the point-slope form, and optionally converting to slope-intercept or standard form. This process is a fundamental skill in algebra and coordinate geometry, with applications in various fields. By understanding the underlying concepts and practicing the steps involved, you can confidently solve similar problems and deepen your understanding of linear relationships. This article has provided a detailed walkthrough of the process, ensuring clarity and comprehension for learners of all levels. The ability to find the equation of a line is not only a mathematical skill but also a tool for problem-solving in real-world scenarios, making it an essential part of mathematical literacy.
