Equation Of A Line Passing Through Two Points (-6, -3) And (6, 6)
Introduction
In the realm of mathematics, particularly in coordinate geometry, determining the equation of a line that passes through two given points is a fundamental concept. This article delves into the process of finding the equation of a line that traverses the points (-6, -3) and (6, 6). Understanding this process is crucial for various applications, including graphing linear equations, solving systems of equations, and modeling real-world scenarios with linear relationships. The equation of a line, often expressed in slope-intercept form (y = mx + b) or point-slope form (y - y1 = m(x - x1)), provides a concise algebraic representation of the line's characteristics. By calculating the slope and identifying a point on the line, we can construct its equation. This article will guide you through the steps, offering clear explanations and examples to solidify your understanding. Let's embark on this mathematical journey to unravel the mysteries of linear equations.
Understanding the Basics
Before diving into the solution, let's revisit some fundamental concepts. A straight line is uniquely defined by two points. The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of every point on the line. There are several forms in which the equation of a line can be expressed, each with its own advantages and uses. The two most common forms are:
- Slope-Intercept Form: y = mx + b, where 'm' represents the slope of the line and 'b' represents the y-intercept (the point where the line crosses the y-axis).
- Point-Slope Form: y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a known point on the line.
The slope of a line, 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 any two points on the line. Mathematically, the slope (m) between two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
The y-intercept, denoted by 'b', is the y-coordinate of the point where the line intersects the y-axis. It represents the value of y when x is equal to 0. Understanding these basic concepts is essential for finding the equation of a line. With a clear grasp of these principles, we can confidently tackle the problem at hand and derive the equation of the line passing through the points (-6, -3) and (6, 6).
Step 1: Calculate the Slope
The first step in finding the equation of the line is to determine the slope. Given the points (-6, -3) and (6, 6), we can use the slope formula:
m = (y2 - y1) / (x2 - x1)
Let's identify our coordinates:
- x1 = -6
- y1 = -3
- x2 = 6
- y2 = 6
Substituting these values into the slope formula, we get:
m = (6 - (-3)) / (6 - (-6))
m = (6 + 3) / (6 + 6)
m = 9 / 12
Simplifying the fraction, we find:
m = 3 / 4
Therefore, the slope of the line passing through the points (-6, -3) and (6, 6) is 3/4. The slope, a crucial parameter, tells us the rate at which the line rises or falls as we move along the x-axis. A positive slope, as in this case, indicates that the line rises from left to right. The value of 3/4 means that for every 4 units we move to the right along the x-axis, the line rises 3 units along the y-axis. This information is vital for visualizing the line and understanding its behavior. With the slope calculated, we can proceed to the next step of finding the equation of the line using either the point-slope form or the slope-intercept form.
Step 2: Use the Point-Slope Form
Now that we have calculated the slope (m = 3/4), 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)
where (x1, y1) is a point on the line and 'm' is the slope. We have two points to choose from: (-6, -3) and (6, 6). Let's use the point (-6, -3) as (x1, y1). Substituting the values into the point-slope form, we get:
y - (-3) = (3/4)(x - (-6))
Simplifying the equation:
y + 3 = (3/4)(x + 6)
This is the equation of the line in point-slope form. While this form is a valid representation of the line, it is often useful to convert it into slope-intercept form (y = mx + b) for easier interpretation and graphing. The point-slope form provides a direct way to express the equation of a line when we know a point on the line and its slope. It highlights the relationship between the slope, a specific point, and the general coordinates (x, y) that satisfy the equation. By choosing either of the given points, we can arrive at the same point-slope form equation, which represents the same line. Next, we will convert this equation into slope-intercept form to obtain a more familiar representation of the line.
Step 3: Convert to Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form (y = mx + b), we need to isolate y on one side of the equation. Starting from the point-slope form equation we derived in the previous step:
y + 3 = (3/4)(x + 6)
First, distribute the (3/4) on the right side of the equation:
y + 3 = (3/4)x + (3/4)(6)
y + 3 = (3/4)x + 18/4
Simplify the fraction:
y + 3 = (3/4)x + 9/2
Now, subtract 3 from both sides of the equation to isolate y:
y = (3/4)x + 9/2 - 3
To subtract 3 from 9/2, we need to express 3 as a fraction with a denominator of 2:
y = (3/4)x + 9/2 - 6/2
Subtract the fractions:
y = (3/4)x + 3/2
Therefore, the equation of the line in slope-intercept form is:
y = (3/4)x + 1.5
This form clearly shows the slope (m = 3/4) and the y-intercept (b = 3/2 or 1.5). The slope-intercept form is particularly useful because it provides a direct visual interpretation of the line. The coefficient of x, which is 3/4, represents the slope, indicating the steepness and direction of the line. The constant term, 3/2, represents the y-intercept, which is the point where the line crosses the y-axis. This form makes it easy to graph the line and understand its behavior. We have now successfully converted the equation into slope-intercept form, providing a clear and concise representation of the line.
Step 4: Verification
To ensure our equation is correct, we can verify it by plugging in the coordinates of the given points (-6, -3) and (6, 6) into the slope-intercept form equation we derived:
y = (3/4)x + 3/2
Let's start with the point (-6, -3):
- Substitute x = -6 and y = -3 into the equation: -3 = (3/4)(-6) + 3/2 -3 = -18/4 + 3/2 -3 = -9/2 + 3/2 -3 = -6/2 -3 = -3
The equation holds true for the point (-6, -3).
Now, let's verify with the point (6, 6):
- Substitute x = 6 and y = 6 into the equation: 6 = (3/4)(6) + 3/2 6 = 18/4 + 3/2 6 = 9/2 + 3/2 6 = 12/2 6 = 6
The equation also holds true for the point (6, 6). Since the equation satisfies both points, we can confidently conclude that our equation is correct. Verification is a crucial step in any mathematical problem-solving process. It helps to identify any potential errors and ensures that the solution obtained is accurate. By plugging in the original points into the derived equation, we can confirm that the equation represents the line passing through those points. This step reinforces our understanding of the relationship between the equation of a line and its points, and it gives us confidence in the correctness of our solution.
Conclusion
In conclusion, we have successfully found the equation of the line that passes through the points (-6, -3) and (6, 6). We began by calculating the slope using the slope formula, which gave us m = 3/4. We then used the point-slope form of a linear equation to derive an initial equation of the line. Subsequently, we converted this equation into slope-intercept form, obtaining the equation y = (3/4)x + 3/2. Finally, we verified our equation by substituting the coordinates of the given points and confirming that they satisfy the equation. This process demonstrates a clear and systematic approach to finding the equation of a line when two points are given. The ability to determine the equation of a line is a fundamental skill in mathematics with applications in various fields, including physics, engineering, and computer science. Understanding the concepts of slope, intercepts, and different forms of linear equations is essential for solving a wide range of problems. By mastering these techniques, we can effectively model and analyze linear relationships in both theoretical and real-world contexts. This article has provided a comprehensive guide to this process, equipping you with the knowledge and skills to confidently tackle similar problems in the future.
