Finding The Equation Of A Perpendicular Line A Step By Step Guide

Introduction

In mathematics, particularly in coordinate geometry, determining the equation of a line that satisfies specific conditions is a fundamental concept. One common problem involves finding the equation of a line that is perpendicular to a given line and passes through a particular point. This article delves into the process of solving such problems, focusing on a step-by-step approach to find the desired equation. We will explore the underlying principles of linear equations, slopes, and the relationship between perpendicular lines. Understanding these concepts is crucial for success in algebra, calculus, and various applications in science and engineering.

The problem at hand is to find the equation of a line that is perpendicular to the line y = -2x + 1 and contains the point (8, 2). To solve this, we will first identify the slope of the given line. The given equation is in slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. By comparing y = -2x + 1 with the slope-intercept form, we can see that the slope of the given line is -2. The slope is a crucial aspect because the slope of any line perpendicular to this line will be the negative reciprocal of -2. This property stems from the geometric relationship that perpendicular lines form a right angle, and their slopes are inversely related. Next, we calculate the negative reciprocal, which will be the slope of our new line. Using this new slope and the given point (8, 2), we can then use the point-slope form of a linear equation to find the equation of the perpendicular line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. After substituting the values, we'll simplify the equation into slope-intercept form to get the final equation of the line. This methodical approach ensures we arrive at the correct solution, providing a clear understanding of how linear equations and their properties work together to solve geometric problems.

Understanding Slopes and Perpendicular Lines

To successfully tackle the problem of finding the equation of a line perpendicular to y = -2x + 1 and passing through the point (8, 2), it is essential to first have a solid grasp of slopes and the relationship between perpendicular lines. The slope of a line, often denoted by m, measures the steepness and direction of the line. Mathematically, it represents the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. In the slope-intercept form of a linear equation, y = mx + b, the coefficient m directly gives us the slope. For example, in the given line y = -2x + 1, the slope m is -2. This indicates that for every one unit increase in x, the value of y decreases by two units. Understanding the slope's magnitude and sign is critical because it not only tells us how steep the line is but also whether it is increasing (positive slope) or decreasing (negative slope) as we move from left to right.

Now, let's delve into perpendicular lines. Two lines are said to be perpendicular if they intersect at a right angle (90 degrees). A fundamental property of perpendicular lines is that their slopes are negative reciprocals of each other. If a line has a slope m, then a line perpendicular to it will have a slope of -1/m. This relationship is crucial because it allows us to find the slope of a perpendicular line directly from the slope of the given line. In our case, the slope of the given line y = -2x + 1 is -2. Therefore, the slope of a line perpendicular to this line will be the negative reciprocal of -2, which is -1/(-2) = 1/2. This positive slope of 1/2 indicates that the perpendicular line will increase as we move from left to right, which is the opposite direction of the original line. Recognizing and applying this negative reciprocal relationship is a key step in solving problems involving perpendicular lines. It bridges the gap between knowing the slope of one line and determining the slope of a line that forms a right angle with it. Mastering these concepts provides a solid foundation for solving more complex problems in geometry and algebra, making it easier to visualize and manipulate linear equations in various contexts.

Calculating the Slope of the Perpendicular Line

In order to determine the equation of the line perpendicular to y = -2x + 1, the first crucial step is to calculate the slope of this perpendicular line. As we discussed earlier, the relationship between the slopes of perpendicular lines is that they are negative reciprocals of each other. This means that if we know the slope of one line, we can easily find the slope of a line perpendicular to it by taking the negative reciprocal. The given equation, y = -2x + 1, is in slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept. By observing the equation, we can identify that the slope of the given line is -2. This negative slope indicates that the line decreases as x increases, moving downwards from left to right on a graph. Now, to find the slope of the line perpendicular to y = -2x + 1, we need to find the negative reciprocal of -2. The negative reciprocal is calculated by first taking the reciprocal of the number and then changing its sign. The reciprocal of -2 is -1/2. Then, we take the negative of -1/2, which is -(-1/2) = 1/2. Therefore, the slope of the line perpendicular to y = -2x + 1 is 1/2. This positive slope tells us that the perpendicular line will increase as x increases, moving upwards from left to right on a graph. Understanding this calculation is essential because it directly impacts the equation of the perpendicular line we are trying to find. If we were to use the incorrect slope, the resulting line would not be perpendicular to the given line, and the solution would be incorrect. This careful step ensures that we are on the right track to solving the problem accurately.

Using the Point-Slope Form

Now that we have determined the slope of the perpendicular line to be 1/2, the next step is to use this slope along with the given point (8, 2) to find the equation of the line. The point-slope form of a linear equation is a very useful tool for this purpose. The point-slope form is given by the equation y - y1 = m(x - x1), where (x1, y1) is a known point on the line and m is the slope of the line. This form is particularly helpful because it allows us to write the equation of a line directly when we know a point on the line and its slope. In our case, we have the point (8, 2) and the slope m = 1/2. We can substitute these values into the point-slope form to get the equation of the line. Substituting x1 = 8, y1 = 2, and m = 1/2 into the point-slope form, we get: y - 2 = (1/2)(x - 8). This equation represents the line that is perpendicular to y = -2x + 1 and passes through the point (8, 2). However, the equation is currently in point-slope form, which is not the most common form for a final answer. To simplify this equation and express it in a more standard form, such as slope-intercept form (y = mx + b), we need to distribute the slope (1/2) on the right side of the equation and then isolate y. This process involves algebraic manipulation, which we will cover in the next section. Using the point-slope form is a critical step in solving this problem because it directly incorporates the given information—the slope and the point—into an equation that represents the line. Understanding and applying this form correctly is essential for obtaining the correct equation of the perpendicular line.

Converting to Slope-Intercept Form

After applying the point-slope form, we arrived at the equation y - 2 = (1/2)(x - 8). While this equation accurately represents the line perpendicular to y = -2x + 1 and passing through the point (8, 2), it is often more convenient to express the equation in slope-intercept form (y = mx + b). The slope-intercept form is beneficial because it explicitly shows the slope (m) and the y-intercept (b) of the line, making it easier to visualize and interpret the line's properties. To convert the equation from point-slope form to slope-intercept form, we need to perform a few algebraic steps. First, we distribute the slope (1/2) across the terms inside the parentheses on the right side of the equation: y - 2 = (1/2)x - (1/2)(8). Simplifying the multiplication, we get: y - 2 = (1/2)x - 4. Now, our goal is to isolate y on the left side of the equation. To do this, we add 2 to both sides of the equation: y - 2 + 2 = (1/2)x - 4 + 2. This simplifies to: y = (1/2)x - 2. This is the equation of the line in slope-intercept form. We can see that the slope m is 1/2, which we calculated earlier, and the y-intercept b is -2. This means that the line crosses the y-axis at the point (0, -2). Converting to slope-intercept form is a crucial step because it provides a clear and concise representation of the line's characteristics. It allows us to easily identify the slope and y-intercept, which are fundamental properties of a linear equation. By completing this step, we have not only found the equation of the line but also expressed it in a form that is widely used and understood in mathematics.

Final Answer

In conclusion, after systematically working through the problem, we have successfully found the equation of the line that is perpendicular to y = -2x + 1 and contains the point (8, 2). We began by identifying the slope of the given line, which was -2. Then, we used the property that the slopes of perpendicular lines are negative reciprocals to determine that the slope of the perpendicular line is 1/2. Next, we employed the point-slope form of a linear equation, y - y1 = m(x - x1), to incorporate the given point (8, 2) and the calculated slope into an equation representing the line. This gave us the equation y - 2 = (1/2)(x - 8). To express the equation in a more standard and easily interpretable form, we converted it to slope-intercept form, y = mx + b. By distributing the slope and isolating y, we arrived at the final equation: y = (1/2)x - 2. This equation clearly shows that the line has a slope of 1/2 and a y-intercept of -2. Therefore, the final answer is y = (1/2)x - 2. This entire process demonstrates the importance of understanding fundamental concepts such as slopes, perpendicular lines, and the different forms of linear equations. By applying these concepts methodically, we were able to solve the problem accurately and efficiently. This approach not only provides the solution but also reinforces the underlying mathematical principles, which are essential for tackling more complex problems in mathematics and related fields.

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