Equation Of A Line Passing Through A Point With A Given Slope
This article delves into the process of determining the equation of a line when given a specific point it passes through and its slope. This is a fundamental concept in linear algebra and coordinate geometry, essential for various applications in mathematics, physics, engineering, and computer science. We will explore the point-slope form of a linear equation, a powerful tool for constructing the equation of a line when the slope and a point on the line are known. We will analyze the given problem, step-by-step, to identify the correct equation that satisfies the specified conditions. Understanding this concept is crucial for solving a wide range of problems related to linear equations and their graphical representation. This article aims to provide a comprehensive explanation, enabling readers to confidently tackle similar problems in the future. Mastering this concept forms a strong foundation for more advanced topics in mathematics and related fields. The ability to find the equation of a line given a point and slope is a cornerstone of analytical geometry, bridging the gap between algebraic expressions and their geometric interpretations. It's a concept that appears frequently in standardized tests and serves as a building block for understanding more complex mathematical models. Therefore, a thorough understanding of this topic is highly beneficial for students and professionals alike. We will also discuss common mistakes and pitfalls to avoid, ensuring a clear and accurate understanding of the point-slope form and its application.
Understanding the Point-Slope Form
Before we tackle the specific problem, let's revisit the point-slope form of a linear equation. This form provides a direct way to express the equation of a line when we know its slope, often denoted by m, and a point (x₁, y₁) that the line passes through. The point-slope form is given by:
y - y₁ = m(x - x₁)
Here, m represents the slope of the line, (x₁, y₁) represents the coordinates of the given point, and (x, y) represents any general point on the line. The beauty of this form lies in its simplicity and direct applicability. You simply substitute the known values of the slope and the point's coordinates into the equation, and you have the equation of the line in point-slope form. This form is particularly useful because it avoids the need to first calculate the y-intercept, which is required in the slope-intercept form (y = mx + b). The point-slope form directly incorporates the given information, making it a more efficient method in many cases. Understanding the derivation of the point-slope form is also beneficial. It stems from the definition of slope as the change in y divided by the change in x. By rearranging this definition algebraically, we arrive at the point-slope form. This connection to the fundamental definition of slope reinforces the understanding of the equation and its geometric interpretation. Furthermore, the point-slope form serves as a bridge to other forms of linear equations, such as the slope-intercept form and the standard form. By manipulating the point-slope form algebraically, we can easily convert it into these other forms, providing flexibility in representing and analyzing linear relationships.
Applying the Point-Slope Form to the Problem
Now, let's apply the point-slope form to solve the given problem. We are given that the line passes through the point (2, -1/2) and has a slope of 3. Therefore, we have:
- x₁ = 2
- y₁ = -1/2
- m = 3
Substituting these values into the point-slope form equation:
y - y₁ = m(x - x₁)
y - (-1/2) = 3(x - 2)
Simplifying the equation, we get:
y + 1/2 = 3(x - 2)
This equation directly matches option C in the given choices. It's important to note the careful substitution of the negative value for y₁. A common mistake is to neglect the negative sign, which would lead to an incorrect equation. The simplification step also highlights the importance of paying attention to algebraic details to arrive at the correct answer. This step-by-step application of the point-slope form demonstrates its effectiveness in finding the equation of a line. By clearly identifying the given information and substituting it into the formula, we can easily derive the equation in a straightforward manner. The resulting equation, y + 1/2 = 3(x - 2), represents a line that satisfies both the given point and the slope condition. This reinforces the understanding that a linear equation is uniquely determined by its slope and a point it passes through.
Analyzing the Answer Choices
Let's analyze the given answer choices to confirm our solution and understand why the other options are incorrect.
A. y - 2 = 3(x + 1/2)
This equation is incorrect because it uses the y-coordinate of the point as a positive value instead of substituting it correctly into the point-slope form. This equation also incorrectly adds 1/2 to x instead of subtracting 2. This equation would represent a line with a slope of 3 passing through the point (-1/2, 2), which is not the given point.
B. y - 3 = 2(x + 1/2)
This equation is incorrect because it swaps the slope and the x-coordinate difference. The slope is given as 3, but this equation uses 2 as the slope. Also, it uses 3 as the y-coordinate of the point, which is not correct. This equation represents a line with a slope of 2 passing through the point (-1/2, 3), which does not match the given conditions.
C. y + 1/2 = 3(x - 2)
This is the correct equation, as we derived in the previous section. It correctly substitutes the given point (2, -1/2) and the slope 3 into the point-slope form. This equation accurately represents the line that passes through the given point and has the specified slope. We can further confirm this by plugging in the point (2, -1/2) into the equation and verifying that it holds true. Substituting x = 2 and y = -1/2 into the equation yields -1/2 + 1/2 = 3(2 - 2), which simplifies to 0 = 0, confirming that the point lies on the line.
D. y + 1/2 = 2(x - 3)
This equation is incorrect because it uses the wrong slope. It uses a slope of 2 instead of the given slope of 3. While it correctly incorporates the y-coordinate of the point, the incorrect slope makes this equation invalid. This equation represents a line with a slope of 2 passing through the point (3, -1/2), which does not match the given slope.
By systematically analyzing each answer choice, we can clearly see why option C is the only correct equation. This process reinforces the importance of careful substitution and attention to detail when working with linear equations.
Conclusion
In conclusion, the correct equation representing a line that passes through (2, -1/2) and has a slope of 3 is y + 1/2 = 3(x - 2). This was determined by applying the point-slope form of a linear equation and carefully substituting the given values. Understanding the point-slope form is crucial for solving problems of this type, and analyzing the incorrect answer choices helps to solidify the understanding of the concept. This skill is fundamental in algebra and is widely applicable in various mathematical and scientific contexts. Mastering the point-slope form allows for efficient and accurate determination of linear equations, which is a valuable asset in problem-solving. Furthermore, this exercise demonstrates the importance of careful attention to detail when working with algebraic equations. A simple mistake in substitution or simplification can lead to an incorrect answer. Therefore, it is essential to practice these concepts and develop a systematic approach to solving linear equation problems. The ability to confidently and accurately find the equation of a line given a point and slope is a cornerstone of mathematical proficiency.
