Parallel Line Equation A Geometry Class Problem

In the realm of geometry, determining the equation of a line that is parallel to another given line is a fundamental concept. This task requires a solid understanding of slope, intercepts, and the various forms of linear equations. In this particular scenario, a geometry class is faced with the challenge of finding the equation of a line that not only runs parallel to the line defined by the equation y - 3 = -(x + 1) but also passes through the specific point (4, 2). Two students, Trish and Demetri, have offered their solutions, presenting us with an opportunity to dissect their approaches and ascertain the correctness of their answers. Trish asserts that the parallel line is represented by the equation y - 2 = -1(x - 4), while Demetri proposes that the parallel line is y = -x + 6. This article aims to delve deep into the process of solving this problem, carefully evaluating both Trish's and Demetri's solutions to determine their validity. We will explore the underlying mathematical principles, meticulously analyze each step, and provide a comprehensive explanation that will not only clarify the correct solution but also enhance your understanding of parallel lines and linear equations. Our discussion will cover the crucial aspects of slope, the point-slope form, and the slope-intercept form, ensuring that you gain a robust grasp of these concepts. By the end of this exploration, you will be equipped with the knowledge and skills to confidently tackle similar problems, making this article an invaluable resource for students, educators, and anyone seeking to deepen their understanding of geometry.

Parallel lines, a cornerstone concept in geometry, are defined as lines that lie in the same plane and never intersect. A crucial characteristic of parallel lines is that they possess the same slope. The slope of a line, often denoted by the letter m, quantifies the steepness and direction of the line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Mathematically, if we have two points (x1, y1) and (x2, y2) on a line, the slope m is given by the formula: m = (y2 - y1) / (x2 - x1). This formula underscores the fundamental relationship between the change in the y-coordinates and the change in the x-coordinates, providing a precise measure of the line's inclination. Understanding the concept of slope is paramount when dealing with parallel lines, as it directly dictates whether two lines will ever intersect. If two lines have the same slope, they are either parallel or coincident (i.e., they are the same line). To determine if parallel lines are distinct, we examine their y-intercepts. The y-intercept is the point where the line crosses the y-axis, and it is represented by the value of y when x = 0. Parallel lines, by definition, have the same slope but different y-intercepts, ensuring that they maintain a constant distance from each other and never meet. This distinct characteristic of parallel lines is essential in various geometric and algebraic applications, including finding equations of lines, solving systems of equations, and analyzing geometric shapes. A thorough grasp of these principles is indispensable for success in geometry and related fields.

To find a line parallel to y - 3 = -(x + 1), the crucial first step is to identify the slope of the given line. The equation y - 3 = -(x + 1) is presented in a form that is not immediately recognizable as the standard slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. Therefore, we must manipulate the equation to transform it into the slope-intercept form. This transformation involves isolating y on one side of the equation. Starting with y - 3 = -(x + 1), we can simplify the right side by distributing the negative sign: y - 3 = -x - 1. Next, we add 3 to both sides of the equation to isolate y: y = -x - 1 + 3. Combining the constants, we obtain the equation in slope-intercept form: y = -x + 2. Now that the equation is in the form y = mx + b, it becomes clear that the coefficient of x, which is -1, is the slope of the line. Thus, the slope m of the given line is -1. Understanding this slope is critical because any line parallel to the given line will have the same slope. This property of parallel lines—having equal slopes—is a fundamental concept in geometry and is the cornerstone for solving problems involving parallel lines. With the slope of the given line now determined, we can proceed to find the equation of the parallel line that passes through the specified point. This involves using the point-slope form of a linear equation, which we will discuss in the subsequent sections.

The point-slope form is a valuable tool for finding the equation of a line when you know a point on the line and its slope. This form is particularly useful in situations like the one we are addressing, where we need to find a line parallel to another and passing through a specific point. The point-slope form is expressed as y - y1 = m(x - x1), where (x1, y1) represents the coordinates of a known point on the line, and m is the slope of the line. In our problem, we are tasked with finding a line parallel to y - 3 = -(x + 1) and passing through the point (4, 2). We have already established that the slope of the given line is -1, and since parallel lines have the same slope, the line we are looking for will also have a slope of -1. Now, we have all the necessary components to use the point-slope form. We substitute the coordinates of the given point (4, 2) into the equation as x1 = 4 and y1 = 2, and we use the slope m = -1. Plugging these values into the point-slope form, we get: y - 2 = -1(x - 4). This equation represents the line that is parallel to the given line and passes through the point (4, 2). The equation y - 2 = -1(x - 4) is a valid form of the equation of the line, and it is precisely the equation that Trish proposed. However, it is often desirable to convert the equation into the slope-intercept form (y = mx + b) for ease of interpretation and comparison. In the next section, we will explore how to convert this equation into slope-intercept form and compare it to Demetri's solution.

Trish's solution, y - 2 = -1(x - 4), is a direct application of the point-slope form of a linear equation, which, as we've discussed, is a perfectly valid method for finding the equation of a line when a point and the slope are known. Trish correctly identified the slope of the parallel line as -1, the same as the given line, and used the point (4, 2) to construct the equation. Her equation, y - 2 = -1(x - 4), accurately represents a line that is parallel to y - 3 = -(x + 1) and passes through the point (4, 2). This demonstrates a solid understanding of the fundamental principles of linear equations and parallel lines. However, to fully assess Trish's solution in the context of the problem, it is essential to consider the different forms in which a linear equation can be expressed. While y - 2 = -1(x - 4) is correct, it is not in the slope-intercept form (y = mx + b), which is often preferred for its clarity in representing the slope and y-intercept of the line. To determine if Trish's solution aligns with Demetri's, we need to convert her equation into slope-intercept form. This involves simplifying the equation and isolating y on one side. Starting with y - 2 = -1(x - 4), we first distribute the -1 on the right side: y - 2 = -x + 4. Then, we add 2 to both sides to isolate y: y = -x + 4 + 2. Combining the constants, we get y = -x + 6. This conversion reveals that Trish's solution, when simplified, is equivalent to the equation y = -x + 6, which is the same as Demetri's solution. Therefore, Trish's initial answer, while in a different form, is indeed correct. This highlights the importance of being able to manipulate and convert linear equations between different forms to facilitate comparison and analysis. In the next section, we will delve into Demetri's solution and further solidify our understanding of the problem.

Demetri's solution, y = -x + 6, presents the equation of the parallel line in slope-intercept form. This form, as we've discussed, is particularly useful because it directly reveals the slope and y-intercept of the line. In the equation y = -x + 6, the coefficient of x, which is -1, represents the slope, and the constant term, 6, represents the y-intercept. To verify the correctness of Demetri's solution, we need to ensure that the line it represents is indeed parallel to the given line y - 3 = -(x + 1) and passes through the point (4, 2). We have already determined that the slope of the given line is -1, and since Demetri's line also has a slope of -1, it satisfies the condition for parallelism. To check if Demetri's line passes through the point (4, 2), we substitute the coordinates x = 4 and y = 2 into the equation y = -x + 6 and see if the equation holds true. Substituting the values, we get: 2 = -(4) + 6. Simplifying the right side, we have: 2 = -4 + 6, which further simplifies to 2 = 2. Since the equation is true, the point (4, 2) lies on the line represented by Demetri's equation. This confirms that Demetri's solution, y = -x + 6, is correct. It accurately represents a line that is parallel to the given line and passes through the specified point. Moreover, as we saw in the analysis of Trish's solution, Demetri's equation is the slope-intercept form of the equation Trish derived using the point-slope form. This equivalence underscores the fact that linear equations can be expressed in different forms while still representing the same line. Demetri's direct approach of providing the equation in slope-intercept form demonstrates a clear understanding of linear equations and their properties. In the final section, we will summarize our findings and provide a comprehensive conclusion to the problem.

In conclusion, both Trish and Demetri have provided correct solutions to the problem of finding the equation of a line parallel to y - 3 = -(x + 1) and passing through the point (4, 2). Trish's solution, y - 2 = -1(x - 4), is expressed in point-slope form, which is a valid and accurate representation of the line. Demetri's solution, y = -x + 6, is expressed in slope-intercept form, which is also correct and provides a clear view of the line's slope and y-intercept. By converting Trish's equation to slope-intercept form, we demonstrated that both solutions are equivalent, further solidifying their correctness. This exercise highlights the importance of understanding different forms of linear equations and the ability to convert between them. The key to solving this problem lies in recognizing that parallel lines have the same slope and applying the point-slope form to find the equation of the line. Both students demonstrated a strong grasp of these concepts, leading them to the correct answer. This detailed analysis provides a comprehensive understanding of the problem and the methods used to solve it, serving as a valuable resource for students and educators alike. The ability to confidently tackle such problems is crucial in the study of geometry and related fields, and this article has aimed to provide the necessary tools and insights for success. Ultimately, the problem underscores the interconnectedness of geometric concepts and the power of algebraic techniques in solving geometric challenges. The successful solutions of Trish and Demetri exemplify the effective application of these principles, providing a clear and concise demonstration of their understanding.