Equation Of Line QR Points Q(0,1) And R(2,7) Explained

Jun 16, 2025 by ADMIN 55 views

Finding the Equation of Line QR: A Comprehensive Guide

In this comprehensive guide, we will delve into the process of determining the equation of a line, specifically focusing on line QR which passes through the points Q(0, 1) and R(2, 7). We will explore the fundamental concepts of linear equations, slope calculation, and point-slope form, ultimately arriving at the correct equation that represents line QR. Understanding how to derive the equation of a line is a crucial skill in mathematics, with applications spanning various fields, including physics, engineering, and computer science. This article aims to provide a clear and detailed explanation, ensuring that readers of all backgrounds can grasp the underlying principles and confidently solve similar problems. We will meticulously walk through each step, providing explanations and justifications to enhance comprehension and build a solid foundation in linear algebra.

Understanding Linear Equations

At the heart of this problem lies the concept of a linear equation, a mathematical expression that describes a straight line on a coordinate plane. Linear equations are characterized by their constant rate of change, often referred to as the slope. The general form of a linear equation is y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept, the point where the line crosses the y-axis. This form, known as the slope-intercept form, provides a direct way to visualize the line's behavior: the slope dictates its steepness and direction, while the y-intercept anchors it on the vertical axis. However, there are other forms of linear equations that are equally useful, such as the point-slope form, which we will utilize extensively in this guide. Understanding the different forms of linear equations and their interrelationships is crucial for solving a wide range of mathematical problems, from simple line plotting to complex system analysis. Each form offers a unique perspective on the line's characteristics, and choosing the appropriate form can significantly simplify the solution process. In the context of our problem, we will leverage the point-slope form to efficiently derive the equation of line QR, given the two points it passes through. This approach allows us to directly incorporate the given information into the equation, leading to a straightforward and accurate solution. The point-slope form, as we will see, is a powerful tool for expressing linear relationships and forms the cornerstone of our analysis.

Calculating the Slope

The slope of a line is a measure of its steepness and direction. It quantifies how much the y-coordinate changes for every unit change in the x-coordinate. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. A slope of zero represents a horizontal line, and an undefined slope corresponds to a vertical line. The slope is a fundamental property of a line and plays a crucial role in determining its equation. To calculate the slope (m) between two points (x1, y1) and (x2, y2), we use the formula: m = (y2 - y1) / (x2 - x1). This formula captures the essence of the slope as the ratio of the vertical change (rise) to the horizontal change (run). In our specific problem, we are given the points Q(0, 1) and R(2, 7). Applying the slope formula, we can substitute the coordinates: m = (7 - 1) / (2 - 0) = 6 / 2 = 3. Therefore, the slope of line QR is 3. This means that for every one unit increase in the x-coordinate, the y-coordinate increases by three units. The calculated slope of 3 is a crucial piece of information that we will use in the subsequent steps to determine the equation of the line. Understanding the concept of slope and its calculation is essential for working with linear equations and forms the basis for many mathematical and scientific applications. The slope not only describes the line's steepness but also its orientation in the coordinate plane, providing a complete picture of its directional characteristics.

Utilizing the Point-Slope Form

The point-slope form of a linear equation is a powerful tool for expressing the equation of a line when we know a point on the line and its slope. 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. This form is particularly useful because it directly incorporates the given information into the equation, making it a convenient starting point for finding the equation of a line. Unlike the slope-intercept form, which requires knowing the y-intercept, the point-slope form only needs a single point on the line. This flexibility makes it a versatile option for solving various problems involving linear equations. In our case, we have already calculated the slope of line QR as 3, and we are given two points on the line: Q(0, 1) and R(2, 7). We can choose either point to plug into the point-slope form. Let's use point Q(0, 1). Substituting the values into the point-slope form, we get: y - 1 = 3(x - 0), which simplifies to y - 1 = 3x. This equation represents line QR in point-slope form, directly reflecting the slope and the point Q that lies on the line. The point-slope form provides a clear and concise way to express the linear relationship between x and y, and it can be easily manipulated to obtain other forms of the equation, such as the slope-intercept form or the standard form. The point-slope form highlights the fundamental connection between the slope, a point on the line, and the equation that defines the line's path.

Identifying the Correct Equation

Now that we have derived the equation of line QR in point-slope form as y - 1 = 3x, we can compare it to the given options to identify the correct answer. Option A is y - 1 = 6x, which has the correct form but an incorrect slope. Our calculated slope is 3, not 6, so this option is incorrect. Option B is y - 1 = 3x, which perfectly matches our derived equation. This option has the correct form and the correct slope, making it the correct answer. Option C is y - 7 = 2x - 6. While this equation represents a line, it does not match our calculated slope and does not pass through both points Q(0, 1) and R(2, 7). Substituting the coordinates of Q into this equation yields 1 - 7 = 2(0) - 6, which simplifies to -6 = -6, a true statement. However, substituting the coordinates of R yields 7 - 7 = 2(2) - 6, which simplifies to 0 = -2, a false statement. Therefore, option C is incorrect. Option D is y - 7 = x - 2. This equation also represents a line, but it does not have the correct slope and does not pass through point Q(0, 1). Substituting the coordinates of Q into this equation yields 1 - 7 = 0 - 2, which simplifies to -6 = -2, a false statement. Thus, option D is incorrect. Through this process of elimination and comparison, we have definitively identified option B, y - 1 = 3x, as the correct equation representing line QR. This exercise demonstrates the importance of understanding the point-slope form and how to use it to derive and verify the equation of a line. The ability to accurately identify the equation of a line is a fundamental skill in mathematics with wide-ranging applications.

Conclusion

In conclusion, by meticulously applying the concepts of slope calculation and the point-slope form, we have successfully determined that the equation representing line QR, which passes through points Q(0, 1) and R(2, 7), is y - 1 = 3x. We began by understanding the fundamental nature of linear equations and the significance of the slope. We then calculated the slope of line QR using the formula m = (y2 - y1) / (x2 - x1), arriving at a slope of 3. Next, we utilized the point-slope form of a linear equation, y - y1 = m(x - x1), to derive the equation of the line, substituting the calculated slope and the coordinates of point Q. This resulted in the equation y - 1 = 3x. Finally, we compared our derived equation with the given options, systematically eliminating incorrect choices and confirming that option B, y - 1 = 3x, is the correct representation of line QR. This comprehensive guide has not only provided a step-by-step solution to the specific problem but also reinforced the underlying principles of linear equations, slope, and the point-slope form. These concepts are essential tools in mathematics and have broad applications in various fields. By mastering these techniques, readers can confidently tackle similar problems and deepen their understanding of linear relationships.