Finding The Equation Of A Line That Passes Through A Given Point

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Which equation represents a line that passes through a specific point given the equations $y-5=\frac{1}{2}(x-1)$, $y-\frac{1}{2}=5(x-1)$, $y-1=\frac{1}{2}(x-5)$, and $y-1=5(x-\frac{1}{2})$?

Finding the equation of a line that passes through a specific point is a fundamental concept in mathematics, particularly in the field of coordinate geometry. This article will delve into the process of identifying the correct equation of a line when given a point and a slope, using the point-slope form of a linear equation. We will explore various equations and determine which one accurately represents a line passing through a given point. Understanding this concept is crucial for various applications in mathematics, physics, engineering, and computer graphics. Let's examine the equations provided and dissect how to identify the correct one.

Understanding the Point-Slope Form

The point-slope form is a powerful tool for expressing the equation of a line. It directly incorporates a known point on the line and the slope of the line, making it particularly useful in scenarios where this information is readily available. The general form of the point-slope equation is:

y - y₁ = m(x - x₁)

Where:

  • (x₁, y₁) is a known point on the line.
  • m is the slope of the line.

This form allows us to easily construct the equation of a line if we know a point it passes through and its slope. The point-slope form highlights the relationship between the change in y and the change in x, which is essentially what the slope represents. It provides a clear and intuitive way to represent a linear equation, making it easy to visualize and manipulate. By substituting the coordinates of a known point and the slope into the equation, we can directly obtain the equation of the line. This form is especially beneficial when dealing with problems that involve finding the equation of a line given specific conditions. The point-slope form provides a direct and efficient method for representing and working with linear equations, ensuring accuracy and clarity in mathematical calculations and problem-solving.

Analyzing the Given Equations

We are presented with four equations, each potentially representing a line passing through a specific point. To determine the correct equation, we need to carefully analyze each one and identify the point and slope it represents. The equations are:

  1. y - 5 = (1/2)(x - 1)
  2. y - (1/2) = 5(x - 1)
  3. y - 1 = (1/2)(x - 5)
  4. y - 1 = 5(x - (1/2))

Each equation is in point-slope form, y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. We can extract this information from each equation by comparing it to the general form. For equation 1, the point is (1, 5) and the slope is 1/2. For equation 2, the point is (1, 1/2) and the slope is 5. For equation 3, the point is (5, 1) and the slope is 1/2. Finally, for equation 4, the point is (1/2, 1) and the slope is 5. This detailed analysis allows us to understand the characteristics of each line represented by the equations. By identifying the point and slope, we can visualize the line and compare it to any given conditions or requirements. This method of analysis is crucial for solving problems involving linear equations and coordinate geometry, as it provides a clear and systematic approach to understanding the properties of lines.

Identifying the Correct Equation

The key to identifying the correct equation lies in accurately extracting the point and slope information from each equation. We can then use this information to verify if the line passes through the specified point and has the correct slope. Each equation is in the form y - y₁ = m(x - x₁), which directly gives us the point (x₁, y₁) and the slope m. For instance, in the equation y - 5 = (1/2)(x - 1), the point is (1, 5) and the slope is 1/2. Similarly, for the equation y - (1/2) = 5(x - 1), the point is (1, 1/2) and the slope is 5. The equation y - 1 = (1/2)(x - 5) gives us the point (5, 1) and the slope 1/2, while y - 1 = 5(x - (1/2)) indicates a point (1/2, 1) and a slope of 5. By carefully analyzing these components, we can match the equation to the correct line characteristics. This process requires a clear understanding of the point-slope form and the ability to accurately read the information embedded within the equation. The ability to correctly identify the point and slope is crucial for solving various problems in coordinate geometry, including finding the equation of a line, determining if a point lies on a line, and comparing different linear equations.

Detailed Explanation of Each Option

Let's break down each equation option in detail to understand what they represent and how they differ:

  1. y - 5 = (1/2)(x - 1): This equation represents a line with a slope of 1/2 passing through the point (1, 5). The slope indicates that for every 2 units moved horizontally, the line rises 1 unit vertically. The point (1, 5) is a specific location on the coordinate plane that this line intersects. This equation is in the point-slope form, making it easy to identify these key characteristics.

  2. y - (1/2) = 5(x - 1): This equation describes a line with a slope of 5 passing through the point (1, 1/2). The steeper slope of 5 means the line rises much more rapidly than the previous one, increasing 5 units vertically for every 1 unit moved horizontally. The point (1, 1/2) marks a different location on the coordinate plane where this line exists. Understanding the slope and point helps visualize the line's position and orientation.

  3. y - 1 = (1/2)(x - 5): This equation represents a line with a slope of 1/2 passing through the point (5, 1). It shares the same slope as the first equation, indicating parallel lines, but it passes through a different point, (5, 1). This means the line will have the same inclination but a different vertical position on the graph. Recognizing this similarity and difference is crucial for geometric problem-solving.

  4. y - 1 = 5(x - (1/2)): This equation defines a line with a slope of 5 passing through the point (1/2, 1). It shares the same slope as the second equation, suggesting parallel lines as well, but its point of intersection with the coordinate plane is (1/2, 1). The combination of slope and point uniquely determines the line's position and direction, making each equation distinct and specific. By meticulously examining these details, we can accurately interpret and apply each equation in various mathematical contexts.

Conclusion: Choosing the Correct Equation

In conclusion, to accurately choose the correct equation that represents a line passing through a given point, a thorough understanding of the point-slope form (y - y₁ = m(x - x₁)) is essential. By meticulously extracting the point (x₁, y₁) and the slope m from each equation, we can determine the specific characteristics of the line it represents. Each equation provides a unique set of parameters that define the line's position and orientation on the coordinate plane. The slope dictates the steepness and direction of the line, while the point indicates a specific location the line passes through. By comparing the extracted information with the given conditions or requirements, we can identify the equation that precisely matches the desired criteria. This process involves careful analysis and a clear understanding of the relationship between the equation and the geometric properties of the line. Mastering this skill is crucial for various applications in mathematics, physics, engineering, and computer graphics, where accurately representing lines and their equations is fundamental. Understanding the point-slope form allows for efficient and precise problem-solving, ensuring that the correct equation is selected to represent the line in question. This skill enables us to confidently navigate and solve a wide range of problems involving linear equations and coordinate geometry, making it a cornerstone of mathematical proficiency.