Equation Of A Line Passing Through Two Points Q(0, 1) And R(2, 7)
When dealing with coordinate geometry, a fundamental concept is determining the equation of a line given two points. The equation of a line provides a concise mathematical representation of the line's path on a coordinate plane. This article aims to thoroughly explain how to find the equation of a line, specifically line QR, which passes through the points Q(0, 1) and R(2, 7). Understanding this process is crucial for various mathematical applications, including solving systems of equations, analyzing geometric shapes, and modeling real-world scenarios. The ability to translate geometric information into algebraic equations is a cornerstone of mathematical proficiency. Therefore, we will delve into the necessary steps and formulas, ensuring a clear and comprehensive understanding of the topic. Let’s embark on this journey to master the art of finding the equation of a line. To begin, we need to revisit the basic forms of linear equations. The slope-intercept form, which is y = mx + b, and the point-slope form, which is y - y₁ = m(x - x₁), are two essential forms we will utilize. Here, m represents the slope of the line, and (x₁, y₁) denotes a point on the line. By mastering these forms and their applications, you will be well-equipped to tackle a wide range of problems related to linear equations.
The slope of a line is a measure of its steepness and direction. It indicates how much the y-value changes for every unit change in the x-value. The formula to calculate the slope (m) between two points (x₁, y₁) and (x₂, y₂) is given by:
m = (y₂ - y₁) / (x₂ - x₁)
In our case, the points are Q(0, 1) and R(2, 7). Let's designate Q as (x₁, y₁) and R as (x₂, y₂). Thus, we have:
x₁ = 0, y₁ = 1 x₂ = 2, y₂ = 7
Substituting these values into the slope formula, we get:
m = (7 - 1) / (2 - 0) = 6 / 2 = 3
Therefore, the slope of line QR is 3. This means that for every one unit increase in x, the y-value increases by three units. This calculation is a crucial first step in determining the equation of the line. The slope provides us with essential information about the line's orientation in the coordinate plane. A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line. Understanding the concept of slope is not only essential for finding the equation of a line but also for interpreting linear relationships in various contexts. This understanding forms the bedrock for more advanced topics in algebra and calculus. Furthermore, accurately calculating the slope is paramount as it directly impacts the subsequent steps in deriving the line's equation. An incorrect slope will lead to an incorrect equation, which will, in turn, affect any further calculations or interpretations based on that equation.
Now that we have calculated the slope (m = 3), we can use the point-slope form of a linear equation to determine 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) for this calculation. Substituting the values, we get:
y - 1 = 3(x - 0)
Simplifying this equation, we have:
y - 1 = 3x
This equation represents line QR in point-slope form. The point-slope form is particularly useful because it directly incorporates the slope and the coordinates of a point on the line. This makes it a convenient tool for finding the equation of a line when these two pieces of information are known. Alternatively, we could have used point R(2, 7) in the point-slope form. If we did so, the equation would be:
y - 7 = 3(x - 2)
Expanding this, we get:
y - 7 = 3x - 6
However, this equation can be rearranged to match the simplified form we obtained using point Q. Specifically, by adding 7 to both sides, we get:
y = 3x + 1
And by subtracting 1 from both sides of our original equation (y - 1 = 3x), we also get:
y = 3x + 1
This demonstrates that regardless of which point is used, the final equation of the line remains consistent. This reinforces the idea that a unique line can be defined by its slope and any point lying on it. The ability to manipulate and simplify equations is a key skill in algebra, and this example highlights how different forms of an equation can be equivalent. The flexibility to choose different starting points and manipulate equations to reach a common form underscores the power and versatility of algebraic methods.
While the equation y - 1 = 3x is a valid representation of line QR, it's often helpful to convert it to slope-intercept form (y = mx + b) for easier interpretation. The slope-intercept form makes it immediately clear what the slope (m) and the y-intercept (b) are. To convert our equation to slope-intercept form, we simply need to isolate y on one side of the equation. From the equation:
y - 1 = 3x
We add 1 to both sides:
y = 3x + 1
Now the equation is in slope-intercept form. We can see that the slope (m) is 3 and the y-intercept (b) is 1. The y-intercept is the point where the line crosses the y-axis, which in this case is (0, 1), matching the coordinates of point Q. This conversion process is straightforward but provides significant insight into the line's properties. The slope-intercept form is particularly useful for graphing linear equations and for comparing different lines. For example, lines with the same slope are parallel, and lines with slopes that are negative reciprocals of each other are perpendicular. Understanding these relationships is crucial for solving more complex geometric problems. Furthermore, the slope-intercept form is widely used in various applications, including modeling linear relationships in physics, economics, and other fields. The ability to convert between different forms of linear equations (point-slope, slope-intercept, and standard form) is a valuable skill that enhances mathematical proficiency and problem-solving capabilities. This flexibility allows you to choose the most convenient form for a particular task or to extract specific information from the equation more easily.
Now, let's compare our equation y - 1 = 3x with the given options:
- A. y - 1 = 6x
- B. y - 1 = 3x
- C. y - 7 = 2x - 6
- D. y - 7 = x - 2
Option B, y - 1 = 3x, matches our derived equation. Therefore, option B is the correct answer. Analyzing the other options highlights the importance of accurate calculations. Option A has an incorrect slope (6 instead of 3). To further verify, let's examine options C and D to demonstrate why they are incorrect. Option C, y - 7 = 2x - 6, represents a line with a different slope and y-intercept. To see this, we can rearrange it into slope-intercept form:
y = 2x - 6 + 7 y = 2x + 1
This line has a slope of 2, which is different from the slope of line QR (which is 3). Therefore, option C is incorrect. Similarly, option D, y - 7 = x - 2, can be rearranged into slope-intercept form:
y = x - 2 + 7 y = x + 5
This line has a slope of 1, which is also different from the slope of line QR. Thus, option D is incorrect. This detailed analysis underscores the significance of careful calculation and comparison. It is crucial to not only derive the equation correctly but also to verify it against the given options to ensure accuracy. The process of elimination can be a powerful tool in multiple-choice questions, but it should be based on sound mathematical reasoning rather than guesswork. By systematically analyzing each option and comparing it with the derived equation, we can confidently identify the correct answer and eliminate potential errors.
In conclusion, the equation that represents line QR, which passes through points Q(0, 1) and R(2, 7), is y - 1 = 3x (Option B). This was determined by first calculating the slope of the line using the slope formula and then applying the point-slope form of a linear equation. Converting to slope-intercept form provided further confirmation of our result. This process underscores the fundamental principles of coordinate geometry and the importance of understanding linear equations. Mastering the techniques for finding the equation of a line is essential for various mathematical applications and problem-solving scenarios. The ability to translate geometric information into algebraic equations and vice versa is a critical skill in mathematics. Throughout this article, we have emphasized the importance of accurate calculations, systematic analysis, and the ability to manipulate equations into different forms. These skills are not only valuable for solving this specific problem but also for tackling a wide range of mathematical challenges. The concepts covered here, such as slope, point-slope form, and slope-intercept form, are foundational building blocks for more advanced topics in algebra, calculus, and other areas of mathematics. Therefore, a thorough understanding of these concepts is crucial for continued success in mathematical studies. By practicing these techniques and applying them to various problems, you can strengthen your mathematical skills and develop a deeper appreciation for the elegance and power of mathematical reasoning.
