Finding The Equation Of A Line Perpendicular To Y+1=-3(x-5) Passing Through (4,-6)
In mathematics, determining the equation of a line that satisfies specific conditions is a fundamental skill. This article delves into the process of finding the equation of a line that is perpendicular to a given line and passes through a specified point. This is a common problem in algebra and geometry, and mastering it requires a solid understanding of linear equations, slopes, and the point-slope form. Let's explore the concepts and steps involved in solving this type of problem.
Understanding Perpendicular Lines and Slopes
To effectively tackle the problem of finding the equation of a perpendicular line, it's crucial to first grasp the concept of perpendicularity and its relationship to slopes. Perpendicular lines are lines that intersect at a right angle (90 degrees). The key characteristic that distinguishes perpendicular lines from other intersecting lines lies in their slopes. The slope of a line, often denoted by m, quantifies its steepness and direction. It represents the change in the y-coordinate for every unit change in the x-coordinate. When two lines are perpendicular, their slopes exhibit a unique relationship: they are negative reciprocals of each other.
In mathematical terms, if one line has a slope of m₁, then a line perpendicular to it will have a slope of m₂, where m₂ = -1/m₁. This relationship forms the cornerstone of our approach to finding the equation of a perpendicular line. Understanding this concept is paramount, as it allows us to determine the slope of the perpendicular line directly from the slope of the given line. For instance, if a line has a slope of 2, the slope of a line perpendicular to it would be -1/2. This inverse and sign-change relationship is essential for accurately solving problems involving perpendicular lines. It's also important to remember that horizontal and vertical lines are special cases of perpendicularity. A horizontal line has a slope of 0, and a vertical line has an undefined slope. They are perpendicular to each other, which fits within the rule of negative reciprocals, considering the reciprocal of 0 leads to an undefined value, aligning with the nature of a vertical line's slope. Therefore, a firm grasp of slopes and their behavior in perpendicular lines is the bedrock for the subsequent steps in finding the equation of a line.
Identifying the Slope of the Given Line
The first step in finding the equation of a line perpendicular to a given line is to identify the slope of the given line. The equation of the given line is in point-slope form: y + 1 = -3(x - 5). The point-slope form of a linear equation is expressed as y - y₁ = m(x - x₁), where m represents the slope of the line, and (x₁, y₁) is a point on the line. This form is particularly useful because it directly reveals the slope and a point on the line. Comparing the given equation with the point-slope form, we can easily identify the slope. In the equation y + 1 = -3(x - 5), the slope m is -3. It's important to recognize that the slope is the coefficient of the (x - x₁) term when the equation is in point-slope form. A common mistake is to overlook the negative sign or to confuse the constants within the parentheses with the slope. Once the slope of the given line is correctly identified, we can proceed to determine the slope of the line perpendicular to it. This step is crucial because the slope of the perpendicular line is directly related to the slope of the given line, as they are negative reciprocals of each other. Therefore, a clear understanding of the point-slope form and the ability to extract the slope from it are essential skills in solving this type of problem. This initial step sets the foundation for the subsequent calculations and ultimately leads to finding the desired equation of the perpendicular line.
Determining the Slope of the Perpendicular Line
Having identified the slope of the given line, the next critical step is to determine the slope of the line perpendicular to it. As previously discussed, the slopes of perpendicular lines are negative reciprocals of each other. This relationship provides a direct method for finding the slope of the perpendicular line. If the slope of the given line is m, then the slope of the perpendicular line, denoted as m_perp, is given by m_perp = -1/m. In our case, the slope of the given line is -3. To find the slope of the perpendicular line, we take the negative reciprocal of -3. This involves two operations: first, we take the reciprocal, which is 1/3, and then we change the sign, making it positive. Therefore, the slope of the perpendicular line is m_perp = -1/(-3) = 1/3. It's crucial to accurately apply the negative reciprocal concept to avoid errors in the subsequent steps. A common mistake is to only take the reciprocal or only change the sign, but not both. Another point to consider is the case when the given line has a slope of 0 (a horizontal line) or an undefined slope (a vertical line). If the given line is horizontal, the perpendicular line will be vertical, and vice versa. In such cases, the negative reciprocal rule still applies, but it's important to recognize that the reciprocal of 0 is undefined, and the negative reciprocal of an undefined slope is 0. Once the slope of the perpendicular line is correctly determined, we have a key piece of information needed to construct the equation of the perpendicular line. This slope, along with the given point that the line passes through, will be used in the next step to apply the point-slope form.
Using the Point-Slope Form
With the slope of the perpendicular line determined, the next step is to utilize the point-slope form to construct the equation of the line. The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where m is the slope of the line, and (x₁, y₁) is a point on the line. This form is particularly advantageous when we know the slope of the line and a point it passes through, which is precisely the information we have in this case. We have already calculated the slope of the perpendicular line to be 1/3, and we are given the point (4, -6) through which the line passes. Substituting these values into the point-slope form, we get: y - (-6) = (1/3)(x - 4). It's crucial to correctly substitute the values into the formula, paying close attention to the signs. A common mistake is to mix up the x and y coordinates or to incorrectly handle the negative signs. Once the values are substituted, the equation becomes: y + 6 = (1/3)(x - 4). This equation represents the line that is perpendicular to the given line and passes through the point (4, -6). The point-slope form provides a direct and efficient way to express the equation of a line given its slope and a point. However, the equation is often further simplified into slope-intercept form or standard form for easier interpretation and comparison. Therefore, the next step typically involves simplifying the equation obtained in the point-slope form to one of these standard forms.
Simplifying the Equation
After obtaining the equation in point-slope form, the final step is to simplify it into a more standard form, typically either slope-intercept form or standard form. The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. This form is particularly useful for quickly identifying the slope and y-intercept of the line. To convert the equation from point-slope form to slope-intercept form, we need to isolate y on one side of the equation. Starting with the equation obtained in the previous step, y + 6 = (1/3)(x - 4), we first distribute the 1/3 on the right side: y + 6 = (1/3)x - 4/3. Next, we subtract 6 from both sides of the equation to isolate y: y = (1/3)x - 4/3 - 6. To combine the constants, we need a common denominator, so we rewrite 6 as 18/3: y = (1/3)x - 4/3 - 18/3. Combining the constants, we get: y = (1/3)x - 22/3. This is the equation of the line in slope-intercept form. Alternatively, we can convert the equation to standard form, which is Ax + By = C, where A, B, and C are integers, and A is non-negative. To convert to standard form, we start with the slope-intercept form, y = (1/3)x - 22/3. First, we multiply both sides of the equation by 3 to eliminate the fractions: 3y = x - 22. Then, we rearrange the terms to get the standard form: x - 3y = 22. Both the slope-intercept form and the standard form are valid representations of the equation of the line. The choice of which form to use often depends on the context of the problem or personal preference. By simplifying the equation, we obtain a clear and concise representation of the line that is perpendicular to the given line and passes through the specified point.
Conclusion
In conclusion, finding the equation of a line that is perpendicular to a given line and passes through a specific point involves several key steps. First, it's essential to understand the relationship between the slopes of perpendicular lines, which are negative reciprocals of each other. Identifying the slope of the given line and then calculating the negative reciprocal is crucial. Next, the point-slope form of a linear equation is used to construct the equation of the perpendicular line, utilizing the calculated slope and the given point. Finally, the equation is simplified into either slope-intercept form or standard form for a more conventional representation. Mastering this process requires a solid understanding of linear equations, slopes, and the different forms of linear equations. By following these steps carefully, one can confidently solve problems involving perpendicular lines and their equations. The ability to find the equation of a perpendicular line is a fundamental skill in mathematics, with applications in various fields such as geometry, calculus, and physics. This comprehensive guide provides a clear and detailed explanation of the process, enabling readers to confidently tackle such problems.
