Equation Of A Line Passing Through (-4,-4) And (2,6) Explained
In mathematics, determining the equation of a line is a fundamental concept. When given two points on a line, we can utilize various methods to find the equation that represents the line. This article delves into a step-by-step approach to finding the equation of a line passing through the points (-4, -4) and (2, 6). We will explore the concepts of slope, point-slope form, and slope-intercept form to arrive at the final equation.
Understanding the Fundamentals
Before diving into the calculations, let's establish a solid understanding of the key concepts involved. A line in a two-dimensional plane can be uniquely defined by its slope and a point on the line. The slope, often denoted by 'm', represents the steepness and direction of the line. It is calculated as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. The point-slope form and slope-intercept form are two common ways to express the equation of a line.
The slope-intercept form is written as y = mx + b, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis). This form is particularly useful for visualizing the line's behavior and easily identifying its slope and y-intercept. The point-slope form, on the other hand, is expressed as y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a known point on the line. This form is advantageous when we have a point and the slope but need to determine the equation. Understanding these fundamental concepts is crucial for successfully finding the equation of the line passing through the given points.
Step 1: Calculating the Slope
The first step in finding the equation of the line is to determine its slope. Given two points, (-4, -4) and (2, 6), we can use the slope formula: m = (y2 - y1) / (x2 - x1). Let's designate (-4, -4) as (x1, y1) and (2, 6) as (x2, y2). Plugging these values into the formula, we get: m = (6 - (-4)) / (2 - (-4)). Simplifying the expression, we have: m = (6 + 4) / (2 + 4) = 10 / 6. Reducing the fraction, we find the slope to be m = 5/3. Therefore, the slope of the line passing through the points (-4, -4) and (2, 6) is 5/3. This positive slope indicates that the line is increasing as we move from left to right on the coordinate plane. The magnitude of the slope, 5/3, tells us how steep the line is; for every 3 units we move horizontally, the line rises 5 units vertically. This information will be essential in the next steps as we construct the equation of the line.
Step 2: Using the Point-Slope Form
Now that we have calculated the slope (m = 5/3), we can use the point-slope form to express 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. We can use either of the given points, (-4, -4) or (2, 6), for (x1, y1). Let's use the point (-4, -4). Substituting the values into the point-slope form, we get: y - (-4) = (5/3)(x - (-4)). Simplifying the equation, we have: y + 4 = (5/3)(x + 4). This equation represents the line in point-slope form. While it is a valid representation, it is often desirable to convert it to slope-intercept form (y = mx + b) for easier interpretation and comparison. The point-slope form is particularly useful because it directly incorporates the slope and a point on the line, making it a straightforward way to represent the line's equation. In the next step, we will transform this equation into slope-intercept form to further clarify the line's properties.
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 isolate 'y' on one side of the equation. Starting with the equation y + 4 = (5/3)(x + 4), we first distribute the 5/3 on the right side: y + 4 = (5/3)x + (5/3)(4). This simplifies to: y + 4 = (5/3)x + 20/3. Now, to isolate 'y', we subtract 4 from both sides of the equation: y = (5/3)x + 20/3 - 4. To combine the constant terms, we need a common denominator. Since 4 can be written as 12/3, the equation becomes: y = (5/3)x + 20/3 - 12/3. Combining the fractions, we get: y = (5/3)x + 8/3. This is the equation of the line in slope-intercept form. We can see that the slope (m) is 5/3, as we calculated earlier, and the y-intercept (b) is 8/3. This form provides a clear representation of the line's slope and where it intersects the y-axis. The slope-intercept form is widely used for graphing lines and comparing their properties.
Final Equation
Therefore, the equation of the line that passes through the points (-4, -4) and (2, 6) is y = (5/3)x + 8/3. This equation represents a line with a slope of 5/3 and a y-intercept of 8/3. We arrived at this equation by first calculating the slope using the slope formula, then utilizing the point-slope form with one of the given points, and finally converting the equation to slope-intercept form for clarity and ease of interpretation. The slope-intercept form allows us to quickly visualize the line and understand its behavior. For every 3 units we move horizontally, the line rises 5 units vertically, and it crosses the y-axis at the point (0, 8/3). This equation uniquely defines the line passing through the specified points and can be used to find other points on the line or to analyze its relationship with other lines in the coordinate plane. Understanding how to find the equation of a line is a fundamental skill in algebra and calculus, with applications in various fields such as physics, engineering, and economics.
Alternative Approach: Using Two-Point Form
Another method to find the equation of a line passing through two points is by using the two-point form directly. The two-point form is given by: (y - y1) / (x - x1) = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two given points. Using the points (-4, -4) and (2, 6), we can substitute the values into the formula. Let (-4, -4) be (x1, y1) and (2, 6) be (x2, y2). Substituting these values, we get: (y - (-4)) / (x - (-4)) = (6 - (-4)) / (2 - (-4)). Simplifying the equation, we have: (y + 4) / (x + 4) = (6 + 4) / (2 + 4), which further simplifies to: (y + 4) / (x + 4) = 10 / 6. Reducing the fraction, we get: (y + 4) / (x + 4) = 5/3. Now, we can cross-multiply to eliminate the fractions: 3(y + 4) = 5(x + 4). Distributing the constants on both sides, we have: 3y + 12 = 5x + 20. To convert this to slope-intercept form, we isolate 'y': 3y = 5x + 20 - 12, which simplifies to: 3y = 5x + 8. Finally, dividing both sides by 3, we get: y = (5/3)x + 8/3. This is the same equation we found using the slope-intercept form, confirming the accuracy of our result. The two-point form provides a direct way to find the equation of a line without explicitly calculating the slope separately. It is a useful alternative method, especially when the slope is not immediately required.
Conclusion
In conclusion, we have successfully found the equation of the line passing through the points (-4, -4) and (2, 6) using multiple methods. We first calculated the slope to be 5/3 using the slope formula. Then, we utilized the point-slope form with the calculated slope and one of the given points to represent the equation of the line. Finally, we converted the equation to slope-intercept form, resulting in y = (5/3)x + 8/3. We also explored an alternative approach using the two-point form, which directly yielded the same equation. Understanding these methods is crucial for solving various problems in mathematics and related fields. The slope-intercept form provides a clear understanding of the line's slope and y-intercept, making it a valuable tool for analyzing and visualizing linear relationships. The ability to find the equation of a line given two points is a fundamental skill that serves as a building block for more advanced mathematical concepts. This skill is not only essential in academic settings but also finds applications in real-world scenarios, such as modeling linear trends, predicting outcomes, and solving optimization problems. By mastering these techniques, one can effectively analyze and interpret linear relationships in various contexts.
