Finding The Equation Of Line QR Passing Through Points Q(0, 1) And R(2, 7)
In mathematics, determining the equation of a line is a fundamental concept with numerous applications. In this comprehensive guide, we will delve into the process of finding the equation of a line, specifically focusing on line QR which passes through the points Q(0, 1) and R(2, 7). We will explore the underlying principles, step-by-step methods, and various forms of linear equations, providing a thorough understanding of this essential mathematical skill. This article aims to provide a detailed explanation suitable for students and anyone seeking to enhance their understanding of linear equations and coordinate geometry. Understanding the equation of a line is crucial not only in mathematics but also in various fields such as physics, engineering, and computer science, where linear relationships are frequently encountered and modeled. We will break down the process into manageable steps, ensuring clarity and comprehension. This guide is designed to help you master the techniques required to solve such problems confidently and accurately.
Understanding the Basics: Slope and Point-Slope Form
Before diving into the specifics of line QR, it's crucial to understand the fundamental concepts of slope and the point-slope form of a linear equation. The slope of a line, often denoted as m, measures its steepness and direction. It is defined as the change in the y-coordinate divided by the change in the x-coordinate between any two points on the line. Mathematically, if we have two points (x1, y1) and (x2, y2), the slope m is calculated as:
m = (y2 - y1) / (x2 - x1)
The point-slope form of a linear equation is a convenient way to represent the equation of a line when we know a point on the line and its slope. The point-slope form is given by:
y - y1 = m(x - x1)
Where (x1, y1) is a point on the line and m is the slope. This form is particularly useful because it allows us to write the equation of a line directly from the slope and a single point, without needing to find the y-intercept first. It provides a flexible and efficient way to express linear relationships, making it an essential tool in coordinate geometry. The point-slope form is a stepping stone to understanding other forms of linear equations, such as the slope-intercept form and the standard form, making it a foundational concept in algebra and calculus.
Step-by-Step Solution: Finding the Equation of Line QR
Now, let's apply these concepts to find the equation of line QR, which passes through points Q(0, 1) and R(2, 7). This involves a step-by-step approach, ensuring clarity and accuracy in our calculations. Our primary goal is to use the coordinates of these points to determine the slope and then apply the point-slope form to derive the equation of the line. Each step is crucial for understanding the underlying process and arriving at the correct answer. This methodical approach not only helps in solving this specific problem but also builds a strong foundation for tackling similar problems in the future. Let's break down the process:
Step 1: Calculate the Slope (m)
First, we need to calculate the slope m of line QR using the coordinates of points Q(0, 1) and R(2, 7). Applying the slope formula:
m = (y2 - y1) / (x2 - x1)
Substitute the coordinates of Q(0, 1) as (x1, y1) and R(2, 7) as (x2, y2):
m = (7 - 1) / (2 - 0)
m = 6 / 2
m = 3
Therefore, the slope of line QR is 3. This value indicates the steepness of the line; for every one unit increase in the x-coordinate, the y-coordinate increases by three units. Understanding the slope is crucial for visualizing the line and predicting its behavior. The positive slope indicates that the line is increasing as we move from left to right on the coordinate plane.
Step 2: Use the Point-Slope Form
Next, we use the point-slope form of a linear equation to express the equation of line QR. The point-slope form is given by:
y - y1 = m(x - x1)
We can use either point Q(0, 1) or point R(2, 7) as (x1, y1). Let's use point Q(0, 1) and the slope m = 3:
y - 1 = 3(x - 0)
This equation represents line QR in point-slope form. The choice of using point Q(0, 1) or R(2, 7) is arbitrary; using either point will lead to an equivalent equation. The point-slope form provides a direct way to express the equation of a line using a known point and the slope, making it a valuable tool in coordinate geometry.
Step 3: Simplify the Equation (Optional)
Finally, we can simplify the equation to obtain a more standard form, such as the slope-intercept form (y = mx + b). Starting from the point-slope form:
y - 1 = 3(x - 0)
y - 1 = 3x
y = 3x + 1
This is the slope-intercept form of the equation, where the slope m is 3 and the y-intercept b is 1. This form is particularly useful for quickly identifying the slope and y-intercept of the line. Alternatively, we can rearrange the equation to match one of the given options. For instance, we can keep it in the form:
y - 1 = 3x
This form is also a valid representation of the equation of line QR and matches one of the answer choices provided. Simplifying the equation helps in comparing it with given options and understanding the linear relationship in different contexts.
Analyzing the Answer Choices
Now, let's analyze the given answer choices to determine which equation correctly represents line QR. This involves comparing each option with the equation we derived using the point-slope form and simplified versions. Understanding how to manipulate and compare linear equations is crucial for solving problems efficiently and accurately. Each option presents a slightly different form of the linear equation, and it's important to be able to recognize equivalent forms. Let's examine each option:
A. y - 1 = 6x
This equation suggests a slope of 6, which is not consistent with our calculated slope of 3. Therefore, option A is incorrect. The slope is a critical parameter in defining a line, and any discrepancy in the slope immediately disqualifies the equation.
B. y - 1 = 3x
This equation matches the point-slope form we derived, using point Q(0, 1) and the slope m = 3. Thus, option B is the correct answer. This demonstrates the importance of the point-slope form in directly expressing the equation of a line when a point and slope are known.
C. y - 7 = 2x - 6
This equation can be rearranged to slope-intercept form to check its validity. Adding 7 to both sides gives:
y = 2x + 1
This equation has a slope of 2, which is different from our calculated slope of 3. Therefore, option C is incorrect. Rearranging equations into standard forms helps in easy comparison and verification.
D. y - 7 = x - 2
This equation can also be rearranged to slope-intercept form. Adding 7 to both sides gives:
y = x + 5
This equation has a slope of 1, which is not consistent with our calculated slope of 3. Therefore, option D is incorrect. The ability to manipulate equations and recognize their key parameters is essential for solving such problems.
Alternative Method: Using Slope-Intercept Form Directly
Another approach to finding the equation of line QR involves directly using the slope-intercept form (y = mx + b). This method provides an alternative way to solve the problem and reinforces the understanding of linear equations. The slope-intercept form is particularly useful when the y-intercept is easily identifiable or can be calculated efficiently. This method highlights the versatility of different forms of linear equations and their applicability in various situations.
Step 1: Calculate the Slope (m) (Same as Before)
As we calculated earlier, the slope m of line QR passing through points Q(0, 1) and R(2, 7) is:
m = 3
Step 2: Find the Y-Intercept (b)
The y-intercept is the y-coordinate of the point where the line intersects the y-axis. Since point Q(0, 1) has an x-coordinate of 0, it lies on the y-axis. Therefore, the y-intercept b is 1. In this case, the y-intercept is directly given by the coordinates of one of the points, making the calculation straightforward. If neither point had an x-coordinate of 0, we would substitute the slope and one point into y = mx + b and solve for b.
Step 3: Write the Equation in Slope-Intercept Form
Now we have the slope m = 3 and the y-intercept b = 1. We can write the equation of line QR in slope-intercept form:
y = mx + b
y = 3x + 1
This equation is equivalent to the simplified form we derived earlier and can be rearranged to match option B: y - 1 = 3x. This confirms our earlier result using the point-slope form. The slope-intercept form provides a clear representation of the linear relationship, making it easy to visualize and analyze.
Conclusion
In conclusion, we have successfully found the equation of line QR, which passes through points Q(0, 1) and R(2, 7), using two different methods: the point-slope form and the slope-intercept form. The correct equation representing line QR is y - 1 = 3x, which corresponds to option B. This comprehensive guide has demonstrated the importance of understanding fundamental concepts like slope, point-slope form, and slope-intercept form in solving problems related to linear equations. Mastering these concepts is crucial for success in algebra and related fields. The step-by-step approach outlined in this guide provides a clear and methodical way to tackle such problems. Furthermore, the ability to use different methods and compare results enhances problem-solving skills and provides a deeper understanding of the underlying mathematical principles. By practicing these techniques, you can confidently solve similar problems and apply these concepts in various mathematical and real-world contexts. Understanding linear equations is a cornerstone of mathematical literacy, opening doors to more advanced topics and applications.
