Finding The Equation Of A Line Passing Through G(4, 6) And H(1, 5)

Hey guys! Today, we're diving into a fundamental concept in mathematics: finding the equation of a line. This is a skill that pops up everywhere, from basic algebra to more advanced calculus and even real-world applications like physics and engineering. So, let's break it down step-by-step, using the points G(4, 6) and H(1, 5) as our example. We'll not only find the equation but also understand the why behind each step. Buckle up, and let's get started!

Understanding the Basics: Slope and Point-Slope Form

Before we jump into calculations, let's quickly recap the key concepts. The equation of a line represents all the points that lie on that line. There are several ways to express this equation, but two forms are particularly useful for this problem: slope-intercept form and point-slope form. The slope-intercept form, which you might remember as y = mx + b, where m is the slope and b is the y-intercept. This form is super intuitive because it directly tells you the line's steepness (m) and where it crosses the y-axis (b). However, for finding the equation given two points, the point-slope form is often the more efficient starting point. The point-slope form looks like this: y - y₁ = m(x - x₁), where m is the slope, and (x₁, y₁) is any point on the line. This form is handy because it directly incorporates the coordinates of a known point on the line. To determine the equation of a line, the slope is a crucial concept that quantifies the line's steepness and direction. Mathematically, the slope (m) is defined as the change in y divided by the change in x, often visualized as "rise over run." Given two points, (x₁, y₁) and (x₂, y₂), the slope can be calculated using the formula: m = (y₂ - y₁) / (x₂ - x₁). This formula essentially measures 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 you move from left to right, while a negative slope indicates a decreasing line. A slope of zero means the line is horizontal, and an undefined slope means the line is vertical. Mastering the concept of slope is crucial for understanding linear relationships and their graphical representations.

Step 1: Calculating the Slope (m)

Our first task is to find the slope (m) of the line passing through G(4, 6) and H(1, 5). Remember the formula: m = (y₂ - y₁) / (x₂ - x₁). Let's assign G as point 1 (x₁, y₁) and H as point 2 (x₂, y₂). So, we have x₁ = 4, y₁ = 6, x₂ = 1, and y₂ = 5. Plugging these values into the formula, we get: m = (5 - 6) / (1 - 4) = -1 / -3 = 1/3. Therefore, the slope of our line is 1/3. This means that for every 3 units we move to the right along the x-axis, the line goes up 1 unit along the y-axis. A positive slope, as we've found, confirms that the line is increasing as we move from left to right on the graph. Understanding the slope is crucial because it dictates the line's direction and steepness. A larger slope (in absolute value) means a steeper line, while a smaller slope means a flatter line. The sign of the slope tells us whether the line is increasing (positive) or decreasing (negative). Now that we've successfully calculated the slope, we're one step closer to finding the equation of the line. The calculation of the slope is a fundamental step in linear algebra and coordinate geometry, paving the way for various applications, including determining the equation of a line, analyzing linear relationships, and solving geometric problems. It's a core concept that builds a foundation for more advanced mathematical topics.

Step 2: Using the Point-Slope Form

Now that we've got the slope (m = 1/3), we can use the point-slope form to build our equation. Recall the point-slope form: y - y₁ = m(x - x₁). We can use either point G(4, 6) or H(1, 5) as our (x₁, y₁). Let's use G(4, 6) for now. Plugging in the values, we get: y - 6 = (1/3)(x - 4). This equation represents the line passing through G(4, 6) with a slope of 1/3. But we're not quite done yet! The point-slope form is a great intermediate step, but we usually want to express the equation in standard form or slope-intercept form. To further explain the point-slope form, it's essentially a way of expressing the equation of a line using a known point on the line and its slope. The formula, y - y₁ = m(x - x₁), directly incorporates this information. The (x₁, y₁) represents the coordinates of a known point, and m is the slope. By substituting these values, we create an equation that holds true for all points on the line. This form is particularly useful when you're given a point and a slope, or when you have two points (as in our case) and you've calculated the slope. It's a flexible form that can be easily transformed into other forms, such as slope-intercept form or standard form, depending on the desired representation. The point-slope form provides a clear and concise way to capture the linear relationship defined by a point and a slope.

Step 3: Converting to Standard Form

Let's convert our equation from point-slope form (y - 6 = (1/3)(x - 4)) to standard form. The standard form of a linear equation is Ax + By = C, where A, B, and C are integers, and A is usually positive. First, we'll get rid of the fraction by multiplying both sides of the equation by 3: 3(y - 6) = 3(1/3)(x - 4). This simplifies to 3y - 18 = x - 4. Next, we want to rearrange the equation to match the Ax + By = C format. Let's subtract 3y from both sides and add 4 to both sides: 3y - 18 - 3y + 4 = x - 4 - 3y + 4. This gives us -14 = x - 3y. To make A positive, we'll multiply both sides by -1: 14 = -x + 3y. Finally, rearranging the terms, we get the equation in standard form: -x + 3y = 14. The standard form of a linear equation, Ax + By = C, offers a different perspective compared to the slope-intercept form. While it doesn't directly reveal the slope or y-intercept, it's valuable for several reasons. Firstly, it presents a symmetrical form where x and y are on the same side of the equation, which can be advantageous in certain algebraic manipulations. Secondly, it's particularly useful when dealing with systems of linear equations, as the coefficients A and B are readily available for methods like elimination. Converting to standard form often involves clearing fractions and rearranging terms, as we've demonstrated. The standard form provides a consistent and standardized representation of a linear equation, facilitating comparisons and further analysis. Understanding and being able to convert between different forms of linear equations, including standard form, is a crucial skill in algebra and its applications.

Alternative Approach: Converting to Slope-Intercept Form

Just for kicks, let's also convert our point-slope equation to slope-intercept form (y = mx + b). Starting from y - 6 = (1/3)(x - 4), we'll distribute the 1/3 on the right side: y - 6 = (1/3)x - 4/3. Now, we'll add 6 to both sides to isolate y: y - 6 + 6 = (1/3)x - 4/3 + 6. To add -4/3 and 6, we need a common denominator. 6 can be written as 18/3, so we have: y = (1/3)x - 4/3 + 18/3. This simplifies to y = (1/3)x + 14/3. This is the slope-intercept form of our equation, where the slope (m) is 1/3 and the y-intercept (b) is 14/3. The slope-intercept form, y = mx + b, is arguably the most intuitive form for understanding the behavior of a line. As we discussed earlier, m represents the slope, indicating the line's steepness and direction, while b represents the y-intercept, the point where the line crosses the y-axis. This form directly reveals these key characteristics, making it easy to visualize the line's position and orientation on the coordinate plane. Converting to slope-intercept form involves isolating y on one side of the equation, as we've demonstrated. This form is widely used for graphing lines, comparing slopes and intercepts of different lines, and solving problems involving linear relationships. The slope-intercept form provides a clear and concise representation of a line's defining features.

Conclusion: Mastering Linear Equations

So, guys, we've successfully found the equation of the line passing through G(4, 6) and H(1, 5). We calculated the slope, used the point-slope form, and then converted our equation to both standard form (-x + 3y = 14) and slope-intercept form (y = (1/3)x + 14/3). Remember, the key takeaways here are the concepts of slope, point-slope form, standard form, and slope-intercept form. Understanding these concepts will empower you to tackle a wide range of problems involving linear equations. The ability to find the equation of a line is a foundational skill in mathematics, opening doors to more advanced topics and real-world applications. From analyzing data trends to modeling physical phenomena, linear equations play a vital role in various fields. By mastering the techniques we've covered, you'll be well-equipped to tackle these challenges. Keep practicing, and you'll become a pro at finding equations of lines! This journey through linear equations highlights the interconnectedness of mathematical concepts. Each step builds upon previous knowledge, reinforcing understanding and creating a solid foundation. By actively engaging with the material, asking questions, and practicing regularly, you'll deepen your grasp of linear equations and unlock their potential in solving problems and making connections across different areas of study. Keep exploring, keep learning, and embrace the power of mathematics!