Equation Of Line QR Passing Through Q(0, 1) And R(2, 7)
This article will explore how to find the equation of a line, specifically line QR, given two points it passes through: Q(0, 1) and R(2, 7). We'll delve into the concepts of slope, point-slope form, and slope-intercept form, providing a step-by-step guide to arrive at the correct equation. Understanding how to determine the equation of a line is a fundamental skill in algebra and geometry, with applications extending to various fields like physics, engineering, and computer science. So, let's embark on this mathematical journey and unlock the secrets of linear equations.
Understanding the Fundamentals: Slope and Linear Equations
Before we dive into the specifics of line QR, let's establish a strong foundation by reviewing the core concepts of slope and linear equations. The slope of a line, often denoted by 'm', quantifies its steepness and direction. It essentially tells us how much the line rises (or falls) for every unit change in the horizontal direction. A positive slope indicates an upward trend, a negative slope signifies a downward trend, a zero slope represents a horizontal line, and an undefined slope corresponds to a vertical line. The formula for calculating the slope between two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
Linear equations, on the other hand, are mathematical expressions that describe straight lines. There are several forms of linear equations, each with its own advantages and applications. Two commonly used forms are the point-slope form and the slope-intercept form. The point-slope form is particularly useful when we know a point on the line and its slope. It is expressed as:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope. The slope-intercept form, as the name suggests, highlights the slope and y-intercept of the line. It is written as:
y = mx + b
where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). By understanding these fundamental concepts and formulas, we are well-equipped to tackle the problem of finding the equation of line QR.
Step 1: Calculating the Slope of Line QR
Our first step in determining the equation of line QR is to calculate its slope. We are given two points on the line, Q(0, 1) and R(2, 7). Let's designate Q as (x1, y1) and R as (x2, y2). This means x1 = 0, y1 = 1, x2 = 2, and y2 = 7. We can now apply the slope formula we discussed earlier:
m = (y2 - y1) / (x2 - x1)
Substituting the values, we get:
m = (7 - 1) / (2 - 0)
m = 6 / 2
m = 3
Therefore, the slope of line QR is 3. This positive slope indicates that the line rises as we move from left to right. Understanding the slope is crucial as it forms the foundation for both the point-slope and slope-intercept forms of the linear equation. It tells us the rate of change of y with respect to x, and in this case, for every unit increase in x, y increases by 3 units. This understanding will help us visualize the line and ensure the equation we derive accurately represents its behavior. Now that we have the slope, we can move on to the next step of finding the equation of the line.
Step 2: Using the Point-Slope Form to Find the Equation
Now that we have calculated the slope of line QR to be 3, we can utilize the point-slope form of a linear equation to find the equation of the line. The point-slope form, as we discussed earlier, is given by:
y - y1 = m(x - x1)
We know the slope, m = 3, and we have two points on the line, Q(0, 1) and R(2, 7). We can use either point to substitute into the point-slope form. Let's use point Q(0, 1), where x1 = 0 and y1 = 1. Substituting these values into the equation, we get:
y - 1 = 3(x - 0)
Simplifying the equation, we have:
y - 1 = 3x
This equation, y - 1 = 3x, is one form of the equation of line QR. This form is particularly useful as it directly shows the relationship between y and x based on a specific point (0, 1) and the slope (3). However, it's not the only way to represent the equation. We can also transform this equation into the slope-intercept form or other equivalent forms, depending on the context or the required format. The beauty of the point-slope form lies in its simplicity and directness in incorporating the slope and a known point to define the line. In the next step, we'll explore how this equation relates to the answer choices provided and confirm our solution.
Step 3: Matching the Equation with the Answer Choices
We have derived the equation of line QR in point-slope form as y - 1 = 3x. Now, let's compare this equation with the answer choices provided to identify the correct representation of the line. The answer choices are:
A. y - 1 = 6x B. y - 1 = 3x C. y - 7 = 2x - 6 D. y - 7 = x - 2
By direct comparison, we can see that answer choice B, y - 1 = 3x, matches the equation we derived. This confirms that our calculations and application of the point-slope form were accurate. While we have already found the correct answer, it's a good practice to analyze the other options to understand why they are incorrect. Answer choice A has the wrong slope (6 instead of 3). Answer choices C and D are in a slightly different form, so we would need to manipulate them algebraically to compare them directly. However, we can quickly verify that they don't represent the same line by substituting one of the points, say Q(0, 1), into the equations. For example, in option C, substituting x = 0 and y = 1 gives 1 - 7 = 2(0) - 6, which simplifies to -6 = -6. This is true, so point Q lies on the line represented by option C. However, let's try point R(2, 7). Substituting x = 2 and y = 7 into option C gives 7 - 7 = 2(2) - 6, which simplifies to 0 = -2. This is false, so point R does not lie on the line represented by option C. Therefore, option C is incorrect. A similar process can be used to verify that option D is also incorrect. This exercise highlights the importance of not just finding the correct answer but also understanding why the other options are wrong, solidifying our understanding of the underlying concepts.
Alternative Approach: Using Slope-Intercept Form
While we successfully found the equation of line QR using the point-slope form, it's beneficial to explore alternative methods to enhance our problem-solving skills. Let's use the slope-intercept form (y = mx + b) to solve this problem. We already calculated the slope, m, to be 3. So, our equation will look like:
y = 3x + b
Now, we need to find the y-intercept, b. To do this, we can substitute either point Q(0, 1) or R(2, 7) into this equation. Let's use point Q(0, 1), where x = 0 and y = 1. Substituting these values, we get:
1 = 3(0) + b
Simplifying the equation, we have:
1 = b
So, the y-intercept, b, is 1. Now we can plug the values of m and b back into the slope-intercept form:
y = 3x + 1
This is the equation of line QR in slope-intercept form. Now, we can manipulate this equation to match the point-slope form we found earlier (y - 1 = 3x). Subtracting 1 from both sides of the slope-intercept form, we get:
y - 1 = 3x
This confirms that both methods lead to the same equation, demonstrating the versatility of linear equation forms. Using the slope-intercept form provides a different perspective on the problem, highlighting the importance of the y-intercept. It also reinforces our understanding of how different forms of linear equations can be equivalent and how we can convert between them. This alternative approach not only helps us verify our solution but also strengthens our overall understanding of linear equations.
Conclusion: Mastering Linear Equations
In this article, we successfully determined the equation of line QR passing through points Q(0, 1) and R(2, 7). We explored two primary methods: the point-slope form and the slope-intercept form, demonstrating their effectiveness in solving this type of problem. We calculated the slope of the line to be 3 and then utilized both forms to arrive at the equation y - 1 = 3x, which corresponds to answer choice B. Furthermore, we discussed the importance of understanding the underlying concepts of slope and linear equations, analyzing incorrect answer choices to solidify our comprehension. We also highlighted the ability to convert between different forms of linear equations, showcasing the flexibility and interconnectedness of mathematical concepts. Mastering linear equations is a crucial step in mathematical proficiency, opening doors to more advanced topics and real-world applications. By practicing and exploring different approaches, we can develop a deeper understanding and confidence in tackling linear equation problems.
Through this comprehensive guide, we've not only found the answer to a specific question but also gained valuable insights into the broader realm of linear equations. Remember, the key to success in mathematics lies in understanding the fundamental principles and practicing consistently. So, continue to explore, question, and apply these concepts to various problems, and you'll undoubtedly excel in your mathematical journey.
