Equation Of A Line Passing Through A Point With A Given Slope

Jul 10, 2025 by ADMIN 62 views

Finding the Equation of a Line Given a Point and Slope

In this comprehensive guide, we will delve into the process of determining the equation of a line given a specific point and slope. This is a fundamental concept in coordinate geometry, essential for various mathematical applications. We will meticulously explore the point-slope form, demonstrate how to apply it, and ultimately identify the correct equation from the provided options. Understanding how to derive linear equations from given data points and slopes is a crucial skill in mathematics, particularly in algebra and calculus.

Understanding the Point-Slope Form

To tackle this problem effectively, it's crucial to grasp the point-slope form of a linear equation. This form provides a straightforward method for constructing a line's equation when you know a point on the line extit{((x₁, y₁))} and the slope extit{(m)}. The point-slope form is expressed as:

${y - y₁ = m(x - x₁)}$

Here:

  extit{y} and 	extit{x} are the general coordinates representing any point on the line.

  extit{(x₁, y₁)} represents the coordinates of the specific point that the line passes through.

  extit{m} denotes the slope of the line, indicating its steepness and direction.

The point-slope form is particularly useful because it directly incorporates the given information – a point and the slope – into the equation. It serves as a bridge, connecting the geometric properties of a line (its slope and a point it passes through) with its algebraic representation (the equation). By substituting the known values into this form, we can easily derive the equation of the line. This approach is not only efficient but also conceptually clear, making it a valuable tool in solving linear equation problems.

Applying the Point-Slope Form to Our Problem

In our problem, we're given that the line passes through the point extit{(2, -1/2)} and has a slope of 3. This information directly corresponds to the variables in the point-slope form. We can identify extit{x₁} as 2, extit{y₁} as -1/2, and extit{m} as 3. The process of applying the point-slope form involves substituting these values into the general equation. This substitution is a critical step in translating the given geometric conditions into an algebraic equation. By correctly placing the values, we set the stage for simplifying and finding the equation that matches one of the provided options. The accurate substitution of these values is the foundation upon which the rest of the solution is built.

Substituting the Values

Substituting these values into the point-slope form equation, we get:

${y - (-1/2) = 3(x - 2)}$

This equation is a direct application of the formula, with the given point and slope now integrated into the expression. Notice how the negative sign in front of -1/2 becomes positive when subtracted, which is a common point of attention in such calculations. This substituted equation is the crucial intermediate step that links the problem's premise to its solution. From here, the next step involves simplifying the equation to match one of the given answer choices. The simplification process will primarily involve algebraic manipulation, ensuring that the fundamental equality of the equation is maintained while transforming it into a recognizable form.

Simplifying the Equation

Simplifying the Left Side

The first step in simplifying the equation extit{y - (-1/2) = 3(x - 2)} involves addressing the double negative on the left side. Subtracting a negative number is equivalent to adding its positive counterpart. Thus, extit{y - (-1/2)} simplifies to extit{y + 1/2}. This seemingly small step is important for clarity and to align the equation with the standard forms we are familiar with. Mathematical simplification is not just about reducing terms; it's also about presenting the equation in a way that is easier to understand and manipulate further. This part of the process highlights the importance of paying close attention to details, such as the signs of numbers, as they can significantly impact the outcome.

Distributing on the Right Side

Next, we focus on the right side of the equation, extit{3(x - 2)}. To simplify this, we apply the distributive property, which involves multiplying the term outside the parentheses (3) by each term inside the parentheses (x and -2). This yields extit{3 * x} and extit{3 * -2}, resulting in extit{3x - 6}. The distributive property is a fundamental algebraic rule, and its correct application is crucial for simplifying expressions. This step expands the equation and prepares it for further simplification or comparison with the given options. Understanding and applying the distributive property is a key skill in algebra and is frequently used in solving various types of equations.

The Simplified Equation

After these simplification steps, our equation transforms from extit{y - (-1/2) = 3(x - 2)} to extit{y + 1/2 = 3(x - 2)}. This is a crucial point in our problem-solving process. The equation is now in a simplified form that directly corresponds to one of the provided options. By carefully performing each step of the simplification, we have arrived at a clear and concise equation that represents the line described in the problem. This simplified form makes it straightforward to identify the correct answer choice, demonstrating the power of algebraic manipulation in solving mathematical problems. The final simplified equation encapsulates all the given information – the point and the slope – in a single, easily recognizable form.

Matching with the Given Options

Now that we have simplified the equation to extit{y + 1/2 = 3(x - 2)}, the next step is to compare it with the options provided in the question. This involves a careful side-by-side comparison to identify which of the given equations matches our simplified form. This stage of the problem-solving process emphasizes attention to detail and the ability to recognize equivalent mathematical expressions. Matching the derived equation with the options is not merely about finding an identical match; it also requires understanding that different forms of an equation can represent the same line. Therefore, it's essential to be vigilant and thoroughly check each option.

The Correct Match

By carefully comparing our simplified equation extity + 1/2 = 3(x - 2)} with the given options, we can clearly see that it matches exactly with the third option extit{y + 1/2 = 3(x - 2). This direct match confirms that we have correctly applied the point-slope form and simplified the equation. The identification of the correct match is the culmination of the problem-solving process, validating the steps taken and the understanding of the underlying mathematical principles. This reinforces the importance of a systematic approach to solving problems, where each step builds upon the previous one to reach the final solution.

Why Other Options Are Incorrect

To reinforce understanding, let's briefly examine why the other options are incorrect. The first option, extit{y - 2 = 3(x + 1/2)}, incorrectly substitutes the point and slope into the point-slope form. The second option, extit{y - 3 = 2(x + 1/2)}, uses the wrong slope and point values. Lastly, the fourth option, extit{y + 1/2 = 2(x - 3)}, correctly uses the point but has an incorrect slope. These discrepancies highlight the significance of accurately applying the point-slope form and the importance of double-checking each value before substituting it into the equation. Understanding why certain options are incorrect is as valuable as knowing the correct answer, as it deepens comprehension of the concepts involved.

Conclusion

In conclusion, the equation that represents a line passing through extit{(2, -1/2)} and having a slope of 3 is extit{y + 1/2 = 3(x - 2)}. This was determined by applying the point-slope form of a linear equation, substituting the given point and slope, simplifying the resulting equation, and then matching it with the provided options. This problem underscores the importance of understanding and correctly applying fundamental algebraic concepts, such as the point-slope form and the distributive property. Furthermore, it highlights the need for a methodical approach to problem-solving, where each step is carefully executed and verified. Mastering these skills is essential for success in mathematics and related fields. By working through this problem, we have not only found the correct answer but also reinforced our understanding of linear equations and their properties.