Equation Of A Line Passing Through (-9,-3) With Slope -6

Understanding the Point-Slope Form

When dealing with linear equations, one of the most valuable forms to understand is the point-slope form. This form allows us to directly construct the equation of a line if we know a single point that the line passes through and the slope of the line. The point-slope form is given by:

y - y₁ = m(x - x₁)**

Where:

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

This form is particularly useful because it directly incorporates the two pieces of information we often have or need to find: a point and a slope. Let's delve deeper into how this form works and why it is so effective.

Breaking Down the Point-Slope Form

To truly appreciate the point-slope form, let's break it down into its components and understand how each part contributes to defining the line. The equation y - y₁ = m(x - x₁) might seem abstract at first, but it is built upon fundamental concepts of linear equations. The slope, m, is a critical element. It represents the rate of change of y with respect to x. In simpler terms, it tells us how much y changes for every unit change in x. A steeper line has a larger slope (in magnitude), while a flatter line has a slope closer to zero. A positive slope indicates that the line rises as we move from left to right, and a negative slope indicates that the line falls. The point (x₁, y₁) provides a fixed reference on the line. This point anchors the line in the coordinate plane. Without a specific point, we would only know the line's direction (given by the slope) but not its exact location. By plugging the coordinates of this point into the equation, we ensure that the line passes through it. The variables x and y in the equation represent any general point on the line. The equation essentially describes the relationship that must hold between the x and y coordinates for any point to lie on the line. This relationship is governed by the slope m and the fixed point (x₁, y₁). The beauty of the point-slope form lies in its simplicity and directness. It allows us to write the equation of a line as soon as we know a point and the slope, without needing to perform additional calculations like finding the y-intercept. This makes it a powerful tool for solving various problems in algebra and geometry. For instance, if we are given two points on a line, we can first calculate the slope using the formula m = (y₂ - y₁) / (x₂ - x₁), and then use the point-slope form with either of the given points to write the equation of the line. Similarly, if we know the slope and a point, we can directly plug these values into the point-slope form to get the equation. This versatility makes the point-slope form an indispensable tool in the study of linear equations.

Why the Point-Slope Form Matters

The point-slope form is not just a formula; it's a way of thinking about lines. It connects the geometric idea of a line's slope and a point on the line to an algebraic equation. This connection is crucial for solving problems and understanding linear relationships. The point-slope form is a powerful tool in various mathematical contexts. In calculus, it helps in finding tangent lines to curves. In physics, it can describe the motion of an object with constant velocity. In computer graphics, it's used to draw lines on the screen. The ability to quickly write the equation of a line given a point and a slope is a valuable skill in these fields. Furthermore, the point-slope form provides a stepping stone to other forms of linear equations, such as the slope-intercept form (y = mx + b) and the standard form (Ax + By = C). By manipulating the point-slope form, we can easily convert it into these other forms, depending on the needs of the problem. For example, by distributing the slope and isolating y, we can transform the point-slope form into the slope-intercept form, which explicitly shows the slope and the y-intercept of the line. Understanding the point-slope form enhances our ability to analyze and solve problems involving linear relationships. It's a fundamental concept that underpins many advanced topics in mathematics and its applications. Whether you're a student learning algebra or a professional working in a STEM field, mastering the point-slope form is an investment in your mathematical toolkit.

Applying the Point-Slope Form to the Problem

In this specific problem, we are given a point (-9, -3) and a slope m = -6. Our goal is to find the equation of the line that passes through this point and has this slope. This is a perfect scenario for using the point-slope form. We can directly plug the given values into the formula and see which of the provided options matches our result. The point-slope form is given by:

y - y₁ = m(x - x₁)**

Here, we have:

• (x₁, y₁) = (-9, -3)
• m = -6

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

y - (-3) = -6(x - (-9))**

Simplifying the equation, we have:

y + 3 = -6(x + 9)**

This equation directly corresponds to one of the given options. Let's examine the options to identify the correct one. The given options are:

• A. y - 9 = -6(x - 3)
• B. y + 9 = -6(x + 3)
• C. y - 3 = -6(x - 9)
• D. y + 3 = -6(x + 9)

By comparing our derived equation y + 3 = -6(x + 9) with the options, we can clearly see that option D matches our result. Therefore, option D is the correct equation that represents the line passing through the point (-9, -3) with a slope of -6. This exercise demonstrates the power and efficiency of the point-slope form in solving linear equation problems. By directly applying the formula with the given point and slope, we can quickly arrive at the correct equation without needing to perform extensive manipulations or calculations. This makes the point-slope form a valuable tool for students and professionals alike.

Step-by-Step Solution Process

Let's recap the step-by-step process we used to solve this problem. This will reinforce the application of the point-slope form and provide a clear method for tackling similar problems in the future. The steps are as follows:

1. Identify the given information: The first step is to clearly identify the point and slope provided in the problem. In this case, we have the point (-9, -3) and the slope m = -6.
2. Recall the point-slope form: Remember the point-slope form equation: y - y₁ = m(x - x₁). This is the foundation of our solution.
3. Substitute the values: Substitute the given point (x₁, y₁) = (-9, -3) and slope m = -6 into the point-slope form: y - (-3) = -6(x - (-9)). This step replaces the variables in the formula with the specific values from the problem.
4. Simplify the equation: Simplify the equation by removing the double negatives: y + 3 = -6(x + 9). This makes the equation cleaner and easier to compare with the given options.
5. Compare with the options: Compare the simplified equation with the given options to find the matching equation. In this case, y + 3 = -6(x + 9) matches option D.
6. Select the correct answer: Choose the option that matches the derived equation. Here, option D is the correct answer.

This step-by-step process provides a structured approach to solving problems using the point-slope form. By following these steps, you can confidently tackle similar problems and ensure accuracy in your solutions. The key is to understand the point-slope form, correctly substitute the given values, and simplify the equation to match one of the provided options. This method is efficient and reliable, making it a valuable tool in your mathematical problem-solving toolkit.

Analyzing the Options

In this section, we'll take a closer look at each of the given options and understand why some are incorrect and why the correct option is indeed the solution. This analysis will not only solidify our understanding of the point-slope form but also help us identify common mistakes and avoid them in the future. The options are:

• A. y - 9 = -6(x - 3)
• B. y + 9 = -6(x + 3)
• C. y - 3 = -6(x - 9)
• D. y + 3 = -6(x + 9)

We know that the correct equation should be in the point-slope form: y - y₁ = m(x - x₁), where (x₁, y₁) = (-9, -3) and m = -6. Let's analyze each option:

Option A: y - 9 = -6(x - 3)

This equation has the correct slope m = -6, but the point it represents is (3, 9). This is because the point-slope form uses subtraction, so y - 9 implies y₁ = 9 and x - 3 implies x₁ = 3. This point does not match the given point (-9, -3), so this option is incorrect.

Option B: y + 9 = -6(x + 3)

This equation also has the correct slope m = -6. However, it represents the point (-3, -9). Again, this is because y + 9 can be rewritten as y - (-9), implying y₁ = -9, and x + 3 can be rewritten as x - (-3), implying x₁ = -3. This point does not match the given point (-9, -3), so this option is incorrect as well.

Option C: y - 3 = -6(x - 9)

In this equation, the slope m = -6 is correct. The point represented here is (9, 3). This is because y - 3 implies y₁ = 3 and x - 9 implies x₁ = 9. This point does not match the given point (-9, -3), making this option incorrect.

Option D: y + 3 = -6(x + 9)

This is the correct option. The slope m = -6 is correct, and the point represented is (-9, -3). We can see this because y + 3 can be rewritten as y - (-3), implying y₁ = -3, and x + 9 can be rewritten as x - (-9), implying x₁ = -9. This matches the given point (-9, -3), so this option is correct.

By analyzing each option, we can see how the point-slope form works and how different points and slopes affect the equation. Understanding these nuances is crucial for accurately solving linear equation problems. This detailed analysis reinforces the importance of careful substitution and interpretation of the point-slope form.

Conclusion: Mastering the Point-Slope Form

In conclusion, the equation that represents a line passing through the point (-9, -3) and having a slope of -6 is option D: y + 3 = -6(x + 9). This was determined by applying the point-slope form of a linear equation, which is a fundamental concept in algebra. Mastering the point-slope form is essential for solving various problems related to linear equations and lines in the coordinate plane.

Key Takeaways

Here are some key takeaways from this discussion:

• The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
• To find the equation of a line using the point-slope form, you need a point on the line and the slope of the line.
• Substitute the given point and slope into the point-slope form and simplify the equation.
• Be careful with signs when substituting values into the point-slope form.
• Analyzing the given options and understanding how they relate to the point-slope form can help you identify the correct equation.
• The point-slope form is a versatile tool that can be used to solve a variety of problems in algebra and geometry.

Further Practice

To further enhance your understanding of the point-slope form, try solving similar problems with different points and slopes. Practice converting equations from point-slope form to other forms, such as slope-intercept form and standard form. This will solidify your knowledge and make you more confident in solving linear equation problems.

By mastering the point-slope form, you'll be well-equipped to tackle a wide range of mathematical challenges involving linear equations. This fundamental concept is a building block for more advanced topics in mathematics and its applications, making it a valuable skill to acquire. Remember to practice regularly and apply the point-slope form in various contexts to truly master it. This will not only improve your problem-solving abilities but also deepen your understanding of linear relationships and their representations.