Finding Equation Of Perpendicular Line Point (6 -2) Line Y=-2x+8

In mathematics, particularly in coordinate geometry, the equation of a line is a fundamental concept. Understanding how to determine the equation of a line given certain conditions is crucial for solving a wide range of problems. This article will delve into the process of finding the equation of a line that passes through a specific point and is perpendicular to another given line. We'll explore the underlying principles, step-by-step methods, and provide a detailed explanation to ensure clarity and comprehension. The problem we will tackle involves finding the equation of a line that contains the point (6, -2) and is perpendicular to the line y = -2x + 8. This is a common type of problem in algebra and geometry, and mastering it will significantly enhance your problem-solving abilities. Let's begin by revisiting some essential concepts about lines and their equations. The slope-intercept form of a linear equation, represented as y = mx + b, is a cornerstone in understanding lines. Here, 'm' represents the slope of the line, which indicates its steepness and direction, and 'b' represents the y-intercept, the point where the line crosses the y-axis. The slope is a critical attribute because it determines the line's inclination—a positive slope means the line rises as you move from left to right, a negative slope means it falls, a zero slope indicates a horizontal line, and an undefined slope signifies a vertical line. The y-intercept is equally important, as it pinpoints exactly where the line intersects the vertical axis, giving us a fixed point to anchor the line's position in the coordinate plane. The relationship between the slopes of perpendicular lines is also a key concept. Two lines are perpendicular if and only if the product of their slopes is -1. This means that if one line has a slope of 'm', the slope of a line perpendicular to it will be '-1/m'. This negative reciprocal relationship is fundamental in finding the equation of a perpendicular line, as we'll see in the solution process. The point-slope form of a linear equation, y - y1 = m(x - x1), is another invaluable tool. This form is especially useful when we know a point (x1, y1) on the line and the slope 'm'. By substituting these values into the point-slope form, we can easily derive the equation of the line. This form bypasses the need to find the y-intercept directly, which can sometimes be cumbersome. It provides a straightforward method to express the equation of a line when we have a point and a slope. In the subsequent sections, we will apply these concepts to solve the given problem, breaking down each step to make the process clear and accessible. By understanding these principles and methods, you'll be well-equipped to tackle similar problems and deepen your understanding of linear equations.

1. Determine the Slope of the Given Line

The first step in finding the equation of a line perpendicular to y = -2x + 8 is to identify the slope of the given line. The slope of a line is a critical piece of information, as it tells us the steepness and direction of the line. The slope-intercept form of a linear equation, which is y = mx + b, provides an easy way to determine the slope. In this form, 'm' represents the slope, and 'b' represents the y-intercept, where the line crosses the y-axis. When the equation of a line is presented in the slope-intercept form, it's straightforward to extract the slope by simply looking at the coefficient of 'x'. This coefficient is the value that multiplies 'x', and it directly corresponds to the slope of the line. In our case, the given line is y = -2x + 8. By comparing this equation to the slope-intercept form y = mx + b, we can identify that the coefficient of 'x' is -2. Therefore, the slope of the given line is -2. This means that for every unit increase in 'x', the value of 'y' decreases by 2. The negative sign indicates that the line slopes downward from left to right. Understanding the slope of the given line is essential because the slope of a line perpendicular to it has a specific relationship. Two lines are perpendicular if their slopes are negative reciprocals of each other. This means that if one line has a slope 'm', the slope of a line perpendicular to it is '-1/m'. To find the slope of the line perpendicular to our given line, we need to calculate the negative reciprocal of -2. The reciprocal of -2 is -1/2, and the negative of -1/2 is 1/2. Therefore, the slope of the line perpendicular to y = -2x + 8 is 1/2. This means the perpendicular line rises as you move from left to right, with a gentler slope compared to the original line. Now that we have the slope of the perpendicular line, we can move on to the next step, which involves using this slope along with the given point (6, -2) to find the equation of the line. The combination of the slope and a point on the line is sufficient information to determine the unique equation of the line. We will use the point-slope form of a linear equation, which is particularly useful in situations where we have a point and a slope. This form allows us to construct the equation of the line without needing to first find the y-intercept. By knowing the slope and a point, we can substitute these values into the point-slope form and then rearrange the equation into a more familiar form, such as slope-intercept form or standard form. This process will be detailed in the subsequent sections, ensuring a clear and step-by-step approach to solving the problem.

2. Determine the Slope of the Perpendicular Line

Now that we've identified the slope of the given line as -2, the next critical step is to determine the slope of the line that is perpendicular to it. Perpendicular lines have a unique relationship in terms of their slopes: they are negative reciprocals of each other. This means that if a line has a slope of m, a line perpendicular to it will have a slope of -1/m. This relationship is fundamental in coordinate geometry and is essential for solving problems involving perpendicular lines. Understanding this principle allows us to easily find the slope of a perpendicular line once we know the slope of the original line. In our case, the given line has a slope of -2. To find the slope of the perpendicular line, we need to calculate the negative reciprocal of -2. The reciprocal of a number is simply 1 divided by that number. So, the reciprocal of -2 is -1/2. The