Equation Of A Line Perpendicular To Y=-2x+2 Passing Through (0,1)
In mathematics, determining the equation of a line is a fundamental concept, particularly when dealing with linear equations. This article delves into the process of finding the equation of a straight line that satisfies two specific conditions: it passes through the point (0, 1) and is perpendicular to the line y = -2x + 2. This problem combines the concepts of slope, intercepts, and perpendicular lines, offering a comprehensive understanding of linear equations. Let's explore the steps involved in solving this problem.
Understanding the Fundamentals
Before we dive into the solution, it's essential to grasp the basic concepts of linear equations and perpendicular lines. A linear equation can be represented in the slope-intercept form, which is y = mx + c, where m is the slope and c is the y-intercept. The slope m indicates the steepness of the line, while the y-intercept c is the point where the line crosses the y-axis. Understanding these components is crucial for manipulating and solving linear equations.
Perpendicular lines, on the other hand, are lines that intersect at a right angle (90 degrees). A key property of perpendicular lines is that their slopes are negative reciprocals of each other. 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 finding the equation of a line perpendicular to a given line.
The Slope-Intercept Form: y = mx + c
The slope-intercept form is a cornerstone in understanding linear equations. The slope (m) provides vital information about the line's inclination. A positive slope indicates an upward slant, while a negative slope indicates a downward slant. The magnitude of the slope reflects the steepness of the line; a larger absolute value means a steeper line. The y-intercept (c) is the point where the line intersects the y-axis, providing a fixed point that helps anchor the line's position on the coordinate plane. The slope-intercept form is not just a formula but a visual representation of a line’s behavior and position, making it easier to analyze and compare different linear relationships.
Perpendicular Lines and Negative Reciprocals
The concept of perpendicularity is crucial in geometry and linear algebra. When two lines intersect at a 90-degree angle, they are said to be perpendicular. The relationship between their slopes is elegantly simple: they are negative reciprocals of each other. This means that if one line has a slope of m, the perpendicular line will have a slope of -1/m. This property stems from the geometric conditions that define perpendicularity. The product of the slopes of two perpendicular lines is always -1, a mathematical testament to their orthogonal relationship. Understanding this concept allows us to deduce the slope of a perpendicular line directly from the slope of the original line, making it a powerful tool in geometric problem-solving.
Step-by-Step Solution
Now, let's apply these concepts to find the equation of the line that passes through the point (0, 1) and is perpendicular to the line y = -2x + 2. The solution involves a series of logical steps, each building upon the previous one.
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Identify the slope of the given line: The given line is in the slope-intercept form y = -2x + 2. By comparing this with the general form y = mx + c, we can see that the slope of the given line (m₁) is -2. This is the first crucial piece of information we need.
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Determine the slope of the perpendicular line: Since the line we want to find is perpendicular to the given line, its slope (m₂) will be the negative reciprocal of m₁. Thus, m₂ = -1/m₁ = -1/(-2) = 1/2. This step is vital as it gives us the direction of the line we are seeking.
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Use the point-slope form: The point-slope form of a linear equation is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. We know that the line passes through the point (0, 1), so x₁ = 0 and y₁ = 1. We also know that the slope of the perpendicular line (m₂) is 1/2. Substituting these values into the point-slope form, we get y - 1 = (1/2)(x - 0).
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Simplify to slope-intercept form: To get the equation in the familiar slope-intercept form, we simplify the equation from the previous step. y - 1 = (1/2)x can be rearranged to y = (1/2)x + 1. This is the equation of the line that passes through (0, 1) and is perpendicular to y = -2x + 2.
Identifying the Slope of the Given Line
The ability to extract the slope from a linear equation is a fundamental skill in algebra. In the slope-intercept form y = mx + c, the coefficient m directly represents the slope of the line. This coefficient quantifies the rate at which the y-value changes for each unit change in the x-value. In our given equation, y = -2x + 2, 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 y-value decreases by 2 units. Recognizing the slope immediately from the equation allows us to proceed with further analysis, such as finding the slope of a perpendicular line or determining the line's steepness and direction.
Determining the Slope of the Perpendicular Line
The relationship between the slopes of perpendicular lines is a key concept in coordinate geometry. As previously mentioned, the slopes of perpendicular lines are negative reciprocals of each other. This mathematical relationship stems from the geometric condition that perpendicular lines intersect at a right angle (90 degrees). If a line has a slope of m, a line perpendicular to it will have a slope of -1/m. This reciprocal and sign-change ensures that the new line forms a 90-degree angle with the original line. In our problem, the slope of the given line is -2. Therefore, the slope of the perpendicular line is -1/(-2) = 1/2. This calculated slope is essential for constructing the equation of the line we seek, as it dictates the direction and steepness of the perpendicular line relative to the original line.
Using the Point-Slope Form
The point-slope form of a linear equation is a powerful tool when you know a point on the line and its slope. The formula y - y₁ = m(x - x₁) directly incorporates this information, where (x₁, y₁) is a known point on the line, and m is the slope. This form is particularly useful because it avoids the need to calculate the y-intercept directly, which can sometimes be cumbersome. By substituting the known point and slope into the formula, we can construct an equation that represents the line. This equation can then be manipulated into other forms, such as slope-intercept or standard form, depending on the context and desired representation. In our problem, we know that the line passes through the point (0, 1) and has a slope of 1/2. Plugging these values into the point-slope form gives us y - 1 = (1/2)(x - 0), a critical step in solving the problem.
Simplifying to Slope-Intercept Form
The slope-intercept form, y = mx + c, is not only the most intuitive but also the most versatile representation of a linear equation. It directly showcases the line's slope (m) and y-intercept (c), making it easy to visualize and compare different lines. Converting an equation into slope-intercept form often involves algebraic simplification, such as distributing terms, combining like terms, and isolating y on one side of the equation. This process not only reveals the slope and y-intercept but also ensures the equation is in its most standard and easily understood form. In our case, starting from the point-slope form y - 1 = (1/2)(x - 0), we simplify it by distributing the 1/2 across the parenthesis, yielding y - 1 = (1/2)x. Adding 1 to both sides of the equation then isolates y, giving us y = (1/2)x + 1. This final equation explicitly shows that the line has a slope of 1/2 and a y-intercept of 1, completing our solution and providing a clear understanding of the line’s properties.
Conclusion
In conclusion, we successfully found the equation of the straight line passing through the point (0, 1) and perpendicular to the line y = -2x + 2. The equation of the line is y = (1/2)x + 1. This problem illustrates the application of fundamental concepts in linear algebra, including the slope-intercept form, the relationship between slopes of perpendicular lines, and the point-slope form. Mastering these concepts is crucial for solving a wide range of mathematical problems involving linear equations. By following the step-by-step solution outlined in this article, you can confidently tackle similar problems and deepen your understanding of linear equations and their applications.
This exercise not only reinforces the mathematical skills required to manipulate linear equations but also enhances the ability to apply these skills in various contexts. The problem demonstrates how abstract concepts such as slope and perpendicularity translate into concrete algebraic expressions, providing a bridge between theoretical knowledge and practical application. As you continue to explore mathematics, remember that each problem solved is a step further in mastering the art of mathematical thinking and problem-solving.
