Finding The Equation Of A Line Given A Point And Slope

In mathematics, especially in coordinate geometry, determining the equation of a line is a fundamental concept. A line can be defined in several ways, one of which is by specifying a point it passes through and its slope. The slope-point form is a powerful tool for finding the equation of a line when you know these two pieces of information. In this article, we will delve into how to use the point-slope form to find the equation of a line, providing a step-by-step explanation and illustrating the concept with an example. Specifically, we will address the problem of finding the equation of a line that passes through the point $(-2, -4)$ with a slope of -6. This article aims to provide a comprehensive guide for students and anyone interested in understanding linear equations.

Understanding the Point-Slope Form

To effectively determine the equation of a line, the point-slope form serves as an indispensable tool. The point-slope form is a specific format used to express the equation of a line, especially when you know a point on the line and its slope. The formula for the point-slope form is given by:

$\qquad y - y_1 = m(x - x_1)$

Where:

• $(x_1, y_1)$ are the coordinates of a known point on the line,
• $m$ is the slope of the line.

This form is particularly useful because it directly incorporates the slope and a point on the line, making it straightforward to write the equation. The slope, denoted as '$m$', represents the steepness and direction of the line. It is calculated as the change in $y$ divided by the change in $x$ between two points on the line. The point $(x_1, y_1)$ is any specific point that the line passes through. By substituting the values of the slope and the coordinates of the point into the point-slope form, we can easily derive the equation of the line. This form highlights the relationship between the line's slope, a specific point on the line, and the general variables $x$ and $y$, which represent any point on the line. Understanding the point-slope form is crucial for solving various problems related to linear equations, as it provides a direct method to express the equation of a line based on its geometric properties. This form not only simplifies the process of finding the equation but also enhances the understanding of the relationship between the slope, points, and the linear equation itself. By mastering the point-slope form, one can efficiently tackle problems involving lines and their equations, making it a fundamental concept in coordinate geometry.

Applying the Point-Slope Form: A Step-by-Step Guide

Now, let's apply the point-slope form to solve the problem at hand: finding the equation of the line that passes through the point $(-2, -4)$ with a slope of -6. This step-by-step guide will walk you through the process, ensuring you understand each step and why it is necessary.

Step 1: Identify the Given Information

The first crucial step is to clearly identify the information provided in the problem. We are given:

• A point on the line: $(-2, -4)$. This means $x_1 = -2$ and $y_1 = -4$.
• The slope of the line: $m = -6$.

Step 2: Write Down the Point-Slope Form

Next, write down the general form of the point-slope equation. This serves as the template for our solution:

$\qquad y - y_1 = m(x - x_1)$

Step 3: Substitute the Given Values

Now, substitute the values of $x_1$, $y_1$, and $m$ into the point-slope form:

$\qquad y - (-4) = -6(x - (-2))$

This substitution is the heart of the process, as it applies the specific information to the general equation, tailoring it to our particular line.

Step 4: Simplify the Equation

The final step is to simplify the equation. Start by dealing with the negative signs:

$\qquad y + 4 = -6(x + 2)$

This simplified equation is the equation of the line in point-slope form. It directly represents the line that passes through the point $(-2, -4)$ with a slope of -6. While this form is perfectly valid, it's often useful to further simplify the equation into slope-intercept form ($y = mx + b$) or standard form ($Ax + By = C$) depending on the context or the requirements of the problem. However, the point-slope form itself provides a clear and concise representation of the line based on the given information. By following these steps, you can confidently find the equation of a line when given a point and a slope, solidifying your understanding of linear equations and their representations. Each step is crucial, from correctly identifying the given information to meticulously substituting and simplifying, ensuring the final equation accurately describes the line.

Comparing with the Given Options

After deriving the equation of the line using the point-slope form, the next step is to compare our result with the given options to identify the correct answer. This involves a careful examination of each option to see which one matches the equation we found.

Our derived equation is:

$\qquad y + 4 = -6(x + 2)$

Now, let's consider the given options:

A. $y + 4 = -6(x + 4)$ B. $y + 2 = -6(x + 2)$ C. $y + 2 = 6(x + 4)$ D. $y + 4 = -6(x + 2)$

By comparing our derived equation with the options, we can see that:

• Option A has an incorrect term inside the parenthesis on the right side of the equation. It should be $(x + 2)$, but it is $(x + 4)$.
• Option B has incorrect terms on the left side of the equation. It should be $y + 4$, but it is $y + 2$.
• Option C has an incorrect slope and incorrect terms. The slope should be -6, but it is 6, and the terms are not consistent with the point $(-2, -4)$.
• Option D, $y + 4 = -6(x + 2)$, perfectly matches our derived equation.

Therefore, the correct answer is Option D. This comparison step is crucial to ensure that the derived equation matches one of the provided options, reinforcing the accuracy of our solution. It also highlights the importance of paying attention to detail and carefully checking each component of the equation. By systematically comparing the derived equation with each option, we can confidently identify the correct answer and avoid potential errors. This process not only helps in solving the immediate problem but also strengthens the understanding of linear equations and their various forms. It emphasizes the significance of precision and accuracy in mathematical problem-solving, ensuring that the final answer is both logically sound and mathematically correct.

Conclusion

In this article, we successfully determined the equation of the line that passes through the point $(-2, -4)$ with a slope of -6. We began by introducing the point-slope form, a fundamental tool in coordinate geometry for finding the equation of a line when a point and the slope are known. We then provided a detailed, step-by-step guide on how to apply the point-slope form, emphasizing the importance of correctly identifying the given information, substituting the values into the formula, and simplifying the equation. By following these steps, we derived the equation $y + 4 = -6(x + 2)$.

Next, we compared our derived equation with the given options, systematically analyzing each one to identify the correct match. This comparison step is crucial for ensuring the accuracy of our solution and reinforcing the understanding of linear equations. We found that Option D, $y + 4 = -6(x + 2)$, perfectly matched our derived equation, confirming it as the correct answer. This exercise not only solved the specific problem but also enhanced our understanding of linear equations and their representations.

Understanding and applying the point-slope form is essential for solving a wide range of problems in mathematics and related fields. It provides a direct and efficient method for expressing the equation of a line based on its geometric properties. By mastering this concept, students and professionals can confidently tackle problems involving lines and their equations, making it a valuable skill in various contexts. The ability to derive and interpret linear equations is a cornerstone of mathematical literacy, enabling us to model and analyze real-world phenomena effectively. This article serves as a comprehensive guide to the point-slope form, equipping readers with the knowledge and skills to confidently solve problems involving linear equations.