Finding The Equation Of Line QR A Comprehensive Guide

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In the realm of coordinate geometry, determining the equation of a line given two points is a fundamental skill. This article will delve into the process of finding the equation of a line, specifically line QR, which passes through the points Q(0,1) and R(2,7). We will explore the underlying principles, the step-by-step calculations, and the different forms in which the equation of a line can be represented. Furthermore, we will meticulously analyze the given options to pinpoint the correct equation that accurately describes line QR. This exploration is crucial for students, educators, and anyone seeking a deeper understanding of linear equations and their graphical representations.

Understanding the Fundamentals of Linear Equations

Before we dive into the specifics of line QR, let's first establish a solid understanding of linear equations in general. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In two-dimensional coordinate geometry, a linear equation represents a straight line. The most common form of a linear equation is the slope-intercept form, which is expressed as:

y = mx + b

Where:

  • y represents the vertical coordinate.
  • x represents the horizontal coordinate.
  • m represents the slope of the line.
  • b represents the y-intercept (the point where the line crosses the y-axis).

The slope of a line, often denoted by m, measures the steepness and direction of the line. 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, the slope is calculated as:

m = (y2 - y1) / (x2 - x1)

Where:

  • (x1, y1) and (x2, y2) are the coordinates of two distinct points on the line.

The y-intercept, denoted by b, is the y-coordinate of the point where the line intersects the y-axis. This is the point where the x-coordinate is zero.

Another important form of a linear equation is the point-slope form, which is expressed as:

y - y1 = m(x - x1)

Where:

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

The point-slope form is particularly useful when we have the slope of a line and a point that lies on the line. It allows us to directly construct the equation of the line without having to first find the y-intercept.

Determining the Equation of Line QR

Now that we have reviewed the fundamental concepts of linear equations, let's apply them to the specific problem of finding the equation of line QR. We are given two points on the line: Q(0,1) and R(2,7). Our goal is to determine which of the given options correctly represents the equation of this line.

Step 1: Calculate the Slope

The first step in finding the equation of line QR is to calculate its slope. Using the formula for slope, we have:

m = (y2 - y1) / (x2 - x1)

Substituting the coordinates of points Q(0,1) and R(2,7) into the formula, we get:

m = (7 - 1) / (2 - 0) = 6 / 2 = 3

Therefore, the slope of line QR is 3. This means that for every 1 unit increase in the x-coordinate, the y-coordinate increases by 3 units.

Step 2: Use the Point-Slope Form

Now that we have the slope, we can use the point-slope form of a linear equation to construct the equation of line QR. We can use either point Q(0,1) or point R(2,7) as our known point. Let's use point Q(0,1). The point-slope form is:

y - y1 = m(x - x1)

Substituting the slope m = 3 and the coordinates of point Q(0,1) into the equation, we get:

y - 1 = 3(x - 0)

Simplifying the equation, we have:

y - 1 = 3x

This is the equation of line QR in point-slope form. Notice that this equation matches option B in the given options.

Step 3: Verify with the Slope-Intercept Form (Optional)

To further confirm our result, we can convert the equation from point-slope form to slope-intercept form. The slope-intercept form is:

y = mx + b

Starting with the equation in point-slope form:

y - 1 = 3x

Add 1 to both sides of the equation:

y = 3x + 1

This equation is now in slope-intercept form. We can see that the slope m is 3 and the y-intercept b is 1. This confirms that the line passes through the point (0,1), which is point Q, and has a slope of 3, which we calculated earlier.

Step 4: Check the Other Options

Although we have found a matching equation, it's crucial to check the other options to ensure that they are incorrect. This reinforces our understanding of why the correct option is indeed the correct one.

  • Option A: y - 1 = 6x

    This equation has a slope of 6, which is not the slope we calculated for line QR (which is 3). Therefore, this option is incorrect.

  • Option C: y - 7 = 2x - 6

    To analyze this option, we can rewrite it in slope-intercept form. Add 7 to both sides:

    y = 2x + 1

    This equation has a slope of 2 and a y-intercept of 1. The slope does not match the slope of line QR (which is 3), so this option is incorrect.

  • Option D: y - 7 = x - 2

    Similarly, let's rewrite this equation in slope-intercept form. Add 7 to both sides:

    y = x + 5

    This equation has a slope of 1 and a y-intercept of 5. The slope does not match the slope of line QR, so this option is also incorrect.

Conclusion: The Correct Equation

After meticulously calculating the slope, using the point-slope form, and verifying the result in slope-intercept form, we have conclusively determined that the equation representing line QR is:

y - 1 = 3x

This equation corresponds to option B in the given choices. Our analysis has also shown why the other options are incorrect, further solidifying our understanding of linear equations and their applications.

Additional Insights and Applications

The process of finding the equation of a line given two points is a fundamental concept in coordinate geometry with numerous applications in various fields. Understanding this process allows us to model linear relationships, make predictions, and solve real-world problems.

Real-World Applications

Linear equations are used extensively in various disciplines, including:

  • Physics: Describing the motion of objects at a constant velocity.
  • Economics: Modeling supply and demand curves.
  • Engineering: Designing structures and systems with linear components.
  • Computer Graphics: Representing lines and shapes in computer graphics and animations.
  • Data Analysis: Identifying trends and patterns in data sets.

Further Exploration

To deepen your understanding of linear equations, consider exploring the following topics:

  • Parallel and Perpendicular Lines: Learn how the slopes of parallel and perpendicular lines are related.
  • Systems of Linear Equations: Investigate methods for solving systems of linear equations, such as substitution and elimination.
  • Linear Inequalities: Explore how to represent and solve linear inequalities graphically and algebraically.
  • Applications of Linear Equations in Calculus: Discover how linear equations are used to approximate curves and functions in calculus.

By mastering the concepts and techniques discussed in this article, you will be well-equipped to tackle a wide range of problems involving linear equations and their applications.

In summary, finding the equation of a line passing through two given points is a crucial skill in coordinate geometry. By calculating the slope and utilizing the point-slope form, we can effectively determine the equation that represents the line. In the case of line QR passing through points Q(0,1) and R(2,7), the correct equation is y - 1 = 3x, which corresponds to option B. This detailed analysis has not only provided the solution but also reinforced the underlying principles of linear equations, empowering readers with a deeper understanding of this fundamental concept.