Finding Equation Of Parallel Line With X-Intercept -3
In mathematics, determining the equation of a line that satisfies certain conditions is a fundamental skill. This article explores the process of finding the equation of a line that is parallel to a given line and has a specific x-intercept. We will delve into the concepts of slope, parallel lines, and x-intercepts, providing a comprehensive guide to solving this type of problem. Understanding these concepts is crucial for various applications in mathematics, physics, and engineering.
Understanding Parallel Lines and Slopes
When tackling problems involving parallel lines, it's crucial to understand the relationship between their slopes. Parallel lines, by definition, never intersect, and this geometric property translates directly into their algebraic representation. The key concept is that parallel lines have the same slope. The slope of a line, often denoted as 'm', represents its steepness and direction. It quantifies the rate of change in the vertical direction (y-axis) for every unit change in the horizontal direction (x-axis). Mathematically, the slope is calculated as the 'rise over run,' which is the change in y divided by the change in x. When two lines have the same slope, it means they have the same steepness and direction, ensuring they will never converge or diverge, thus remaining parallel.
Consider the given line equation: y = (2/3)x + 3. This equation is in slope-intercept form, which is y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. By comparing this general form with the given equation, we can directly identify the slope of the given line as 2/3. This value is paramount because any line parallel to this given line must also have a slope of 2/3. Understanding this fundamental property is the cornerstone for solving problems involving parallel lines. It allows us to immediately narrow down the possibilities and focus on lines with the correct slope. In essence, the slope acts as a unique identifier for the direction of a line, and parallel lines share this identifier.
Therefore, when searching for a line parallel to y = (2/3)x + 3, we are essentially looking for a line with the same steepness and direction, meaning its equation must also have a slope of 2/3. The y-intercept, however, can be different, leading to a family of parallel lines, each shifted vertically but maintaining the same inclination. This understanding forms the basis for the subsequent steps in solving the problem, where we will use the given x-intercept to pinpoint the specific line within this family that satisfies the given conditions. The ability to extract and utilize the slope from a line's equation is a critical skill in coordinate geometry, enabling us to analyze and manipulate lines based on their directional properties.
Utilizing the X-intercept
The x-intercept of a line is the point where the line crosses the x-axis. At this point, the y-coordinate is always zero. This seemingly simple fact is a powerful tool for determining the equation of a line, especially when combined with the slope. In this problem, we are given that the desired line has an x-intercept of -3. This means the line passes through the point (-3, 0). This single point, along with the slope we previously determined, provides us with enough information to uniquely define the line. The x-intercept acts as an anchor point, fixing the line's position on the coordinate plane.
To effectively utilize the x-intercept, we can employ the point-slope form of a linear equation. The point-slope form is a versatile representation of a line, expressed as y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a point on the line. This form is particularly useful when we know the slope and a point, as is the case in this problem. By substituting the known values into the point-slope form, we can directly construct the equation of the line. The point-slope form elegantly captures the relationship between the slope, a specific point on the line, and the general coordinates (x, y) of any other point on the line.
In our case, we know the slope m = 2/3 (from the parallel line condition) and a point (x1, y1) = (-3, 0) (from the x-intercept). Plugging these values into the point-slope form, we get: y - 0 = (2/3)(x - (-3)). This equation is a direct representation of the line we are seeking. It encapsulates both the slope requirement (parallel to the given line) and the x-intercept condition. The next step involves simplifying this equation into a more familiar form, such as the slope-intercept form, to make it easier to compare with the given options and to further analyze the line's properties. The point-slope form is a bridge, connecting the geometric information (slope and a point) to the algebraic representation (the equation of the line).
Converting to Slope-Intercept Form
To effectively compare our derived equation with the given options and gain a clearer understanding of the line's properties, we need to convert the equation from point-slope form to slope-intercept form. The slope-intercept form, y = mx + b, is a standard representation of a linear equation, where 'm' is the slope and 'b' is the y-intercept. This form is particularly useful because it directly reveals the slope and y-intercept, making it easy to visualize the line's position and orientation on the coordinate plane. Converting to slope-intercept form involves algebraic manipulation to isolate 'y' on one side of the equation.
Starting with the equation obtained from the point-slope form, y - 0 = (2/3)(x - (-3)), we can simplify it step-by-step. First, we simplify the expression inside the parentheses: y = (2/3)(x + 3). Next, we distribute the (2/3) across the terms inside the parentheses: y = (2/3)x + (2/3)(3). Finally, we simplify the last term: y = (2/3)x + 2. This final equation is in slope-intercept form. We can now readily identify the slope as 2/3 (which confirms it's parallel to the given line) and the y-intercept as 2.
This conversion process highlights the power of algebraic manipulation in revealing the underlying properties of a line. By transforming the equation from one form to another, we gain different perspectives and insights. The slope-intercept form, in particular, provides a clear and concise representation of the line's key characteristics. In this case, the equation y = (2/3)x + 2 clearly demonstrates that the line is parallel to the given line (same slope of 2/3) and intersects the y-axis at the point (0, 2). This transformation is a crucial step in solving the problem, allowing us to confidently identify the correct answer from the given options. The ability to convert between different forms of linear equations is a fundamental skill in algebra and coordinate geometry, enabling us to analyze and manipulate lines effectively.
Identifying the Correct Equation
Now that we have derived the equation of the line in slope-intercept form, y = (2/3)x + 2, we can easily compare it with the given options. The equation we found represents a line that is parallel to the given line y = (2/3)x + 3 (since they both have the same slope of 2/3) and has an x-intercept of -3 (as we used this information in our derivation). By systematically applying the concepts of parallel lines, slopes, and x-intercepts, we have successfully pinpointed the unique equation that satisfies the given conditions. The process of elimination and comparison is a crucial step in problem-solving, ensuring that we have arrived at the correct answer.
Examining the other options, we can see why they are incorrect. The equations y = -(3/2)x + 3 and y = -(3/2)x + 2 have a slope of -3/2, which is the negative reciprocal of 2/3. This means these lines are perpendicular, not parallel, to the given line. The equation y = (2/3)x + 3 has the same slope as our desired line but a different y-intercept. This means it is parallel to the given line but does not have the specified x-intercept of -3. By carefully analyzing each option, we can confidently rule out the incorrect ones and affirm our solution.
The correct equation, y = (2/3)x + 2, is the only one that meets both criteria: it is parallel to the given line (same slope) and has an x-intercept of -3. This conclusion reinforces the importance of understanding the fundamental concepts of linear equations and their properties. The ability to manipulate equations, extract key information, and compare them with given conditions is a cornerstone of mathematical problem-solving. In this case, we have demonstrated how to systematically approach a problem involving parallel lines and x-intercepts, leading to a clear and accurate solution. The process of identifying the correct equation is not just about finding the answer; it's about solidifying our understanding of the underlying mathematical principles.
Conclusion
In conclusion, finding the equation of a line parallel to a given line with a specific x-intercept involves understanding the relationship between slopes of parallel lines and utilizing the x-intercept as a point on the line. By applying the point-slope form and converting to slope-intercept form, we can systematically derive the desired equation. This process not only provides the solution but also reinforces fundamental concepts in coordinate geometry, highlighting the importance of algebraic manipulation and logical reasoning in problem-solving.
This exercise demonstrates how mathematical problems can be approached in a structured and logical manner. By breaking down the problem into smaller, manageable steps, we can leverage key concepts and formulas to arrive at the correct solution. The ability to translate geometric conditions into algebraic equations and vice versa is a powerful skill in mathematics, with applications extending far beyond the classroom. The principles discussed in this article serve as a foundation for more advanced topics in linear algebra and calculus, emphasizing the importance of a solid understanding of fundamental concepts.
