Finding The Equation Of A Line Passing Through Two Points (-5,-1) And (5,5)
In mathematics, one of the fundamental concepts is understanding linear equations and their graphical representation as straight lines. Given two points in a Cartesian plane, there exists a unique line that passes through them. The challenge often lies in finding the equation of this line. This article delves into the process of determining the equation of a line that passes through the points (-5, -1) and (5, 5). We will explore the underlying principles, the step-by-step calculations, and the various forms of linear equations. This comprehensive guide is designed to provide a clear understanding of how to approach such problems, making it an invaluable resource for students and enthusiasts alike.
The equation of a line is typically represented in several forms, each with its own advantages and uses. The most common forms are the slope-intercept form, the point-slope form, and the standard form. Understanding these forms is crucial for effectively working with linear equations. The slope-intercept form, y = mx + b, is particularly useful because it directly reveals the slope (m) and the y-intercept (b) of the line. The point-slope form, y - y1 = m(x - x1), is beneficial when you have the slope and a point on the line. Lastly, the standard form, Ax + By = C, is often used for general representations and can simplify certain algebraic manipulations. To find the equation of a line passing through two given points, we will primarily focus on using the slope-intercept and point-slope forms, as they offer a straightforward approach to the problem.
Calculating the Slope
The first step in finding the equation of a line that passes through two points is to determine the slope. The slope, often denoted by m, measures the steepness and direction of the line. It is defined as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. Mathematically, the slope m between two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
This formula is derived from the concept of rise over run, where the rise is the vertical change (y2 - y1) and the run is the horizontal change (x2 - x1). The slope tells us how much the y-value changes for every unit change in the x-value. A positive slope indicates that the line is increasing (going upwards) as we move from left to right, while a negative slope indicates that the line is decreasing (going downwards). A slope of zero means the line is horizontal, and an undefined slope (division by zero) indicates a vertical line.
For the given points (-5, -1) and (5, 5), we can substitute the coordinates into the slope formula. Let's denote (-5, -1) as (x1, y1) and (5, 5) as (x2, y2). Plugging these values into the formula, we get:
m = (5 - (-1)) / (5 - (-5))
Simplifying the expression, we have:
m = (5 + 1) / (5 + 5)
m = 6 / 10
Reducing the fraction, we find the slope:
m = 3 / 5
Thus, the slope of the line that passes through the points (-5, -1) and (5, 5) is 3/5. This positive slope indicates that the line is increasing, which means as the x-value increases, the y-value also increases. Knowing the slope is a critical piece of information, as it allows us to proceed to the next step of finding the equation of the line.
Using the Point-Slope Form
Having calculated the slope, the next step is to use this value along with one of the given points to form the equation of the line. The point-slope form of a linear equation is particularly useful in this scenario. As mentioned earlier, the point-slope form is given by:
y - y1 = m(x - x1)
where m is the slope, and (x1, y1) is a point on the line. This form is derived from the definition of slope and provides a direct way to write the equation of a line when you know its slope and a point it passes through. The point-slope form is advantageous because it minimizes the steps required to find the equation, especially when compared to directly substituting into the slope-intercept form and solving for the y-intercept.
We have already determined that the slope m of the line is 3/5. Now, we can choose either of the given points (-5, -1) or (5, 5) to substitute into the point-slope form. Let's use the point (-5, -1) as (x1, y1). Substituting these values into the point-slope form, we get:
y - (-1) = (3/5)(x - (-5))
Simplifying the equation, we have:
y + 1 = (3/5)(x + 5)
This equation represents the line in point-slope form. While this form is a valid representation of the line, it is often necessary to convert it into slope-intercept form (y = mx + b) to easily identify the y-intercept and to facilitate further analysis or graphing. Converting to slope-intercept form involves distributing the slope and isolating y on one side of the equation.
Converting to Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form, we need to distribute the slope (3/5) and isolate y. Starting from the equation we derived using the point-slope form:
y + 1 = (3/5)(x + 5)
First, distribute the 3/5 across the terms inside the parenthesis:
y + 1 = (3/5)x + (3/5)(5)
Simplifying the multiplication:
y + 1 = (3/5)x + 3
Next, to isolate y, subtract 1 from both sides of the equation:
y = (3/5)x + 3 - 1
Simplifying the subtraction, we get:
y = (3/5)x + 2
This equation is now in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. In this case, the slope m is 3/5, which we calculated earlier, and the y-intercept b is 2. The y-intercept is the point where the line crosses the y-axis, which means the line passes through the point (0, 2).
The slope-intercept form provides a clear and concise representation of the line, making it easy to understand the line's behavior. The slope indicates the rate of change of y with respect to x, and the y-intercept tells us the value of y when x is 0. This form is particularly useful for graphing the line, as we can easily plot the y-intercept and use the slope to find other points on the line. Furthermore, the slope-intercept form is widely used in various mathematical and real-world applications, making it an essential concept to master.
Conclusion and Answer
In summary, we have successfully determined the equation of the line that passes through the points (-5, -1) and (5, 5). The process involved first calculating the slope using the formula m = (y2 - y1) / (x2 - x1), which yielded a slope of 3/5. We then used the point-slope form of a linear equation, y - y1 = m(x - x1), to create an initial equation of the line. By substituting one of the given points and the calculated slope into this form, we obtained y + 1 = (3/5)(x + 5). Finally, we converted this equation to slope-intercept form, y = mx + b, by distributing the slope and isolating y, resulting in the equation y = (3/5)x + 2.
Therefore, the correct answer is:
C. y = (3/5)x + 2
This exercise demonstrates the fundamental principles of linear equations and their applications. Understanding how to find the equation of a line given two points is a crucial skill in mathematics, with applications in various fields such as physics, engineering, and computer science. The ability to manipulate and convert between different forms of linear equations, such as point-slope and slope-intercept forms, enhances problem-solving capabilities and provides a deeper understanding of linear relationships.
By following the steps outlined in this article, one can confidently approach similar problems and accurately determine the equation of a line passing through any two given points. The key takeaway is the systematic approach: calculate the slope, use the point-slope form, and convert to slope-intercept form for a clear and concise representation of the line. This method not only provides the correct answer but also reinforces the underlying concepts of linear equations, ensuring a solid foundation for further mathematical studies.
