Equation Of A Line Passing Through (-6, 7) And (-3, 6)
This comprehensive guide will walk you through the process of determining the equation of a line that passes through two given points: (-6, 7) and (-3, 6). We'll explore the fundamental concepts, formulas, and step-by-step calculations involved in finding the slope and y-intercept, ultimately leading to the equation of the line in slope-intercept form. Understanding these principles is crucial for various mathematical applications and problem-solving scenarios.
Understanding the Fundamentals
Before we delve into the calculations, let's establish the core concepts. A linear equation represents a straight line on a coordinate plane, and its most common form is the slope-intercept form: y = mx + b, where 'm' denotes the slope and 'b' represents the y-intercept. The slope signifies the steepness of the line and its direction (positive or negative), while the y-intercept is the point where the line crosses the vertical y-axis. To find the equation of a line, we need to determine both the slope and the y-intercept using the given points.
Calculating the Slope
The slope, often represented by 'm', quantifies the line's steepness. It's calculated as the change in y divided by the change in x between any two points on the line. Given two points (x1, y1) and (x2, y2), the formula for the slope is: m = (y2 - y1) / (x2 - x1). In our case, the points are (-6, 7) and (-3, 6). Substituting these values into the formula, we get: m = (6 - 7) / (-3 - (-6)) = -1 / 3. Therefore, the slope of the line passing through these points is -1/3. The negative slope indicates that the line slopes downwards from left to right.
Determining the Y-Intercept
The y-intercept, denoted by 'b', is the point where the line intersects the y-axis. To find the y-intercept, we can use the slope-intercept form (y = mx + b) and substitute the slope we just calculated (-1/3) and the coordinates of one of the given points into the equation. Let's use the point (-3, 6). Substituting these values, we get: 6 = (-1/3) * (-3) + b. Simplifying the equation, we have 6 = 1 + b. Subtracting 1 from both sides, we find b = 5. Thus, the y-intercept of the line is 5, meaning the line crosses the y-axis at the point (0, 5).
Forming the Equation
Now that we have the slope (m = -1/3) and the y-intercept (b = 5), we can write the equation of the line in slope-intercept form (y = mx + b). Substituting the values, we get: y = (-1/3)x + 5. This is the equation of the line that passes through the points (-6, 7) and (-3, 6). This equation allows us to determine any point on the line by simply substituting a value for x and solving for y, or vice versa.
To solidify your understanding, let's break down the solution into a step-by-step process:
Step 1: Calculate the Slope
Use the formula m = (y2 - y1) / (x2 - x1) with the given points (-6, 7) and (-3, 6).
- m = (6 - 7) / (-3 - (-6))
- m = -1 / 3
The slope of the line is -1/3.
Step 2: Determine the Y-Intercept
Substitute the slope and one of the points into the slope-intercept form (y = mx + b). Let's use the point (-3, 6).
- 6 = (-1/3) * (-3) + b
- 6 = 1 + b
- b = 5
The y-intercept is 5.
Step 3: Form the Equation
Write the equation in slope-intercept form using the calculated slope and y-intercept.
- y = (-1/3)x + 5
The equation of the line is y = (-1/3)x + 5.
This step-by-step approach provides a clear and concise method for finding the equation of a line when given two points. By following these steps, you can confidently solve similar problems and enhance your understanding of linear equations.
Now, let's examine the given answer choices in the context of our calculated equation, y = (-1/3)x + 5:
A. y = (-1/3)x + 9
This equation has the correct slope (-1/3) but an incorrect y-intercept (9). Therefore, it is not the correct answer. While the slope matches, the line represented by this equation would intersect the y-axis at a different point than the line we are seeking.
B. y = (-1/3)x + 5
This equation matches our calculated equation perfectly. It has the correct slope (-1/3) and the correct y-intercept (5). Thus, this is the correct answer. This equation accurately represents the line that passes through the given points (-6, 7) and (-3, 6).
C. y = -3x - 11y
This equation is not in slope-intercept form and appears to have an error. It contains 'y' on both sides of the equation, making it difficult to directly interpret the slope and y-intercept. Furthermore, if we attempt to rearrange it into slope-intercept form, it will not match our calculated equation. Therefore, it is not the correct answer.
D. y = -3x + 25
This equation has an incorrect slope (-3) and an incorrect y-intercept (25). It does not represent the line that passes through the given points. The steepness of the line and its intersection with the y-axis are significantly different from what we calculated. Therefore, it is not the correct answer.
By carefully analyzing each answer choice and comparing it to our calculated equation, we can confidently identify the correct answer and eliminate the incorrect ones. This process reinforces the importance of understanding the slope-intercept form and accurately calculating the slope and y-intercept.
While we've focused on the slope-intercept form, there are alternative methods to find the equation of a line, each offering a unique perspective and approach. Exploring these methods can deepen your understanding and provide you with versatile problem-solving skills.
Point-Slope Form
The point-slope form of a linear equation is given by: y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a point on the line. This form is particularly useful when you know the slope and a single point on the line. To use this method, first, calculate the slope as we did before (m = -1/3). Then, choose one of the given points, say (-6, 7), and substitute the values into the point-slope form: y - 7 = (-1/3)(x - (-6)). Simplifying this equation, we get: y - 7 = (-1/3)(x + 6). Distributing the -1/3, we have: y - 7 = (-1/3)x - 2. Finally, adding 7 to both sides, we arrive at the slope-intercept form: y = (-1/3)x + 5, which is the same equation we obtained earlier. The point-slope form offers a direct way to incorporate a known point and the slope into the equation.
Standard Form
The standard form of a linear equation is given by: Ax + By = C, where A, B, and C are constants. To convert the slope-intercept form (y = (-1/3)x + 5) to standard form, we first eliminate the fraction by multiplying the entire equation by 3: 3y = -x + 15. Then, we rearrange the terms to get x and y on the same side: x + 3y = 15. This is the equation in standard form. While the standard form doesn't directly reveal the slope and y-intercept, it's a common form used in various mathematical contexts and can be useful for certain types of problems. Converting to standard form demonstrates the flexibility of linear equations and their different representations.
By understanding and utilizing these alternative methods, you can approach linear equation problems from various angles and choose the method that best suits the given information and your problem-solving style. Each form offers a unique way to represent and analyze linear relationships, enhancing your mathematical toolkit.
When working with linear equations, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:
- Incorrectly Calculating the Slope: The slope formula (m = (y2 - y1) / (x2 - x1)) is crucial. Make sure you subtract the y-coordinates and x-coordinates in the correct order. A common mistake is to reverse the order or mix up the coordinates, leading to an incorrect slope. Double-check your calculations and ensure you're using the formula correctly.
- Sign Errors: Pay close attention to signs, especially when dealing with negative numbers. For instance, subtracting a negative number is the same as adding a positive number, and vice versa. A simple sign error can significantly alter the slope or y-intercept, leading to an incorrect equation. Always review your calculations for potential sign errors.
- Incorrectly Substituting Values: When finding the y-intercept or using the point-slope form, ensure you substitute the values of x and y correctly. Using the wrong values will result in an incorrect equation. Take your time and carefully substitute the appropriate values into the equation.
- Forgetting to Simplify: After substituting values and performing calculations, always simplify the equation to its simplest form. This makes it easier to compare your answer with the given choices and ensures you have the correct equation. Simplifying also helps in identifying the slope and y-intercept more clearly.
By being aware of these common mistakes and taking the necessary precautions, you can minimize errors and confidently solve linear equation problems. Careful attention to detail and thorough checking of your work are key to success.
Linear equations aren't just abstract mathematical concepts; they have numerous real-world applications that make them essential tools in various fields. Understanding these applications can help you appreciate the practical significance of linear equations and their relevance to everyday life.
- Modeling Relationships: Linear equations are used to model relationships between two variables that have a constant rate of change. For example, the relationship between the number of hours worked and the amount earned at an hourly wage can be modeled using a linear equation. The slope represents the hourly wage, and the y-intercept represents any fixed amount earned regardless of hours worked. This allows for predicting earnings based on hours worked or vice versa.
- Predicting Trends: Linear equations can be used to predict future trends based on past data. For instance, if you have data on sales figures over several months, you can use a linear equation to model the trend and predict future sales. The slope would represent the rate of change in sales, and the equation can be used to estimate sales for upcoming months. This is a valuable tool in business and finance for forecasting and planning.
- Calculating Distances and Speeds: Linear equations are used in physics to describe motion with constant velocity. For example, the equation distance = rate × time is a linear equation where the rate (speed) is the slope. If you know the speed and time, you can calculate the distance traveled, or if you know the distance and speed, you can calculate the time. This principle is fundamental in navigation, transportation, and various engineering applications.
- Budgeting and Finance: Linear equations are used in budgeting to track income and expenses. A simple linear equation can represent the relationship between income, expenses, and savings. For example, if you have a fixed income and certain fixed expenses, you can use a linear equation to determine how much you can save each month. This helps in managing personal finances effectively.
These are just a few examples of the many real-world applications of linear equations. From modeling relationships to predicting trends, calculating distances, and managing finances, linear equations provide a powerful framework for understanding and solving a wide range of problems. Recognizing these applications enhances your understanding of the practical value of linear equations.
In summary, we have successfully determined the equation of the line that passes through the points (-6, 7) and (-3, 6). By calculating the slope (-1/3) and y-intercept (5), we arrived at the equation y = (-1/3)x + 5. This process involved understanding the slope-intercept form, applying the slope formula, and substituting values to find the y-intercept. We also explored alternative methods like the point-slope form and standard form, providing a comprehensive understanding of linear equations. Furthermore, we discussed common mistakes to avoid and highlighted the numerous real-world applications of linear equations. Mastering these concepts and techniques will equip you with valuable problem-solving skills in mathematics and beyond. Linear equations are fundamental tools for modeling, predicting, and analyzing relationships in various fields, making their understanding essential for both academic and practical pursuits.
