What Is The Ordered Pair For The Point On The X-axis That Lies On The Line Parallel To The Given Line And Passes Through The Point (-6, 10)?

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In this article, we will delve into the process of finding the ordered pair for a point on the x-axis that lies on a line parallel to a given line and passing through a specific point. This is a fundamental concept in coordinate geometry, often encountered in algebra and pre-calculus courses. We'll break down the problem step-by-step, ensuring a clear understanding of the underlying principles and calculations involved. Understanding these concepts is crucial for various applications in mathematics, physics, and engineering, where the ability to determine the relationship between lines and points is essential for solving complex problems. This article will not only guide you through the solution but also enhance your overall problem-solving skills in coordinate geometry. Let's embark on this journey to master the art of finding ordered pairs on the x-axis parallel to a given line.

Problem Statement

To illustrate the process, let's consider the specific problem of finding the ordered pair for the point on the x-axis that lies on the line parallel to a given line and passes through the point (-6, 10). This type of problem typically involves several steps, including determining the slope of the given line, finding the equation of the parallel line, and identifying the x-intercept of the parallel line. Each of these steps requires a solid understanding of basic algebraic concepts and geometric principles. Before diving into the detailed solution, it's important to understand the context and the significance of each step. We'll first review the basic concepts of slopes and parallel lines, followed by a step-by-step approach to solving the problem. This will ensure that you not only understand the solution but also grasp the underlying principles, enabling you to tackle similar problems with confidence. This introduction sets the stage for a comprehensive exploration of the problem, preparing you for the detailed explanation that follows.

Understanding Parallel Lines and Slopes

Parallel lines are lines in the same plane that never intersect. A key characteristic of parallel lines is that they have the same slope. The slope of a line 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. The slope is a crucial parameter in determining the characteristics of a line, and it plays a significant role in identifying parallel lines. Understanding the concept of slope is fundamental not only in coordinate geometry but also in various other mathematical and scientific disciplines. For instance, in physics, the slope of a velocity-time graph represents acceleration. In economics, the slope of a supply or demand curve can provide insights into market behavior. Therefore, a thorough understanding of slopes and their properties is essential for a wide range of applications. In the context of this problem, the slope of the given line will be the same as the slope of the parallel line we are trying to find. This key insight will guide us in determining the equation of the parallel line, which is a critical step in finding the desired ordered pair on the x-axis. The next section will delve into the steps involved in finding the equation of the parallel line and the x-intercept.

Steps to Solve the Problem

To find the ordered pair on the x-axis, we need to follow a series of steps:

  1. Determine the slope of the given line: The problem does not explicitly provide the equation of the original line. We'll assume there's missing information and proceed generally. If we had the equation in slope-intercept form (y = mx + b), the slope would be m. Without the original equation, we can't find a numerical value for the slope yet. However, we'll represent it as m for now and continue the process conceptually. This step is crucial because the slope of the parallel line will be the same as the slope of the original line. Understanding this principle is essential for solving problems involving parallel lines. The slope represents the steepness and direction of the line, and since parallel lines have the same steepness and direction, their slopes must be equal. In practical terms, this means that for every unit increase in the x-coordinate, the y-coordinate changes by the same amount for both lines. This property allows us to use the slope of the given line to determine the equation of the parallel line, which is a key step in finding the ordered pair on the x-axis. The next step will involve using this slope and the given point to find the equation of the parallel line.

  2. Find the equation of the parallel line: Since the parallel line has the same slope (m) and passes through the point (-6, 10), we can use the point-slope form of a line, which is y - y₁ = m(x - x₁). Plugging in the point (-6, 10), we get y - 10 = m(x + 6). This is the equation of the line parallel to the given line and passing through the point (-6, 10). The point-slope form is a powerful tool for finding the equation of a line when you know its slope and a point it passes through. It's derived directly from the definition of slope and provides a straightforward way to express the relationship between the x and y coordinates of points on the line. Understanding the point-slope form is crucial not only for solving problems involving parallel lines but also for a wide range of applications in coordinate geometry and calculus. In this case, using the point-slope form allows us to express the equation of the parallel line in terms of the slope m and the given point (-6, 10). The next step will involve converting this equation into slope-intercept form to make it easier to find the x-intercept.

  3. Find the x-intercept of the parallel line: The x-intercept is the point where the line crosses the x-axis. At this point, the y-coordinate is 0. To find the x-intercept, we set y = 0 in the equation of the parallel line and solve for x. So, we have 0 - 10 = m(x + 6), which simplifies to -10 = m(x + 6). Now, we need to isolate x. Dividing both sides by m, we get -10/m = x + 6. Subtracting 6 from both sides, we find x = -10/m - 6. This is the x-coordinate of the x-intercept. The x-intercept is a crucial point on the line, as it represents the value of x when y is zero. Finding the x-intercept is a common task in coordinate geometry and is often used to graph lines and solve systems of equations. In practical terms, the x-intercept can represent a variety of real-world scenarios, such as the break-even point in business or the initial condition in a physical system. In this case, the x-intercept represents the point where the parallel line intersects the x-axis, which is the ordered pair we are trying to find. The final step will be to express this x-coordinate as an ordered pair.

  4. Express the x-intercept as an ordered pair: The x-intercept is the point where the line crosses the x-axis, so the y-coordinate is 0. The ordered pair is therefore (-10/m - 6, 0). This is the general form of the ordered pair for the x-intercept of the parallel line. The ordered pair represents a specific location on the coordinate plane and is a fundamental concept in coordinate geometry. It consists of two values, the x-coordinate and the y-coordinate, which uniquely identify a point. In this case, the ordered pair (-10/m - 6, 0) represents the point where the parallel line intersects the x-axis. This point is crucial because it satisfies the condition of being on the x-axis and lying on the line parallel to the given line and passing through the given point. However, without a specific value for m (the slope), we cannot determine the exact numerical coordinates of this point. This highlights the importance of having complete information in a problem, as missing information can lead to a general solution rather than a specific one. In the next section, we'll discuss how to handle scenarios where the slope m is known and how to arrive at a numerical solution.

Addressing the Missing Slope Information

In the given problem statement, the equation of the original line is not provided, which means we cannot determine the numerical value of the slope (m). To complete the problem and match one of the answer choices (A, B, C, or D), we need to assume there was a typo or missing information in the original question. Let's analyze the answer choices and work backward to see if we can infer a possible slope for the original line.

The answer choices are:

A. (6, 0) B. (0, 6) C. (-5, 0) D. (0, -5)

We are looking for an ordered pair in the form (x, 0) since it's on the x-axis. This narrows our choices to A (6, 0) and C (-5, 0). Let's consider each possibility:

Case 1: If the x-intercept is (6, 0)

If the x-intercept of the parallel line is (6, 0), then we can plug this into our equation for the x-coordinate: 6 = -10/m - 6. Adding 6 to both sides gives 12 = -10/m. Multiplying both sides by m gives 12m = -10. Dividing both sides by 12 gives m = -10/12 = -5/6. So, if the slope of the parallel line (and the original line) is -5/6, then the x-intercept would be (6, 0).

Case 2: If the x-intercept is (-5, 0)

If the x-intercept of the parallel line is (-5, 0), then we can plug this into our equation for the x-coordinate: -5 = -10/m - 6. Adding 6 to both sides gives 1 = -10/m. Multiplying both sides by m gives m = -10. So, if the slope of the parallel line (and the original line) is -10, then the x-intercept would be (-5, 0).

Determining the Correct Answer Choice

Based on our analysis, we have two possible scenarios:

  • If the slope of the original line is -5/6, the ordered pair on the x-axis is (6, 0).
  • If the slope of the original line is -10, the ordered pair on the x-axis is (-5, 0).

Without the original equation of the line, we cannot definitively determine the correct answer. However, by working backward from the answer choices, we have identified two possible slopes that would lead to the given x-intercepts. If we had the original equation, we could calculate the slope and compare it to these possibilities. This process highlights the importance of having all the necessary information in a problem and demonstrates how we can use logical reasoning and algebraic manipulation to arrive at a solution even when some information is missing. In the next section, we'll discuss the importance of verifying the solution and how to check the answer in similar problems.

Conclusion

In conclusion, to find the ordered pair on the x-axis that lies on a line parallel to a given line and passes through a specific point, we follow these steps: determine the slope of the given line, find the equation of the parallel line using the point-slope form, find the x-intercept by setting y = 0, and express the x-intercept as an ordered pair. However, in this particular problem, the lack of the original line's equation prevented us from finding a definitive numerical answer. We demonstrated how to work backward from the answer choices to infer possible slopes, highlighting the importance of complete information in problem-solving. This exercise underscores the fundamental principles of coordinate geometry, including the properties of parallel lines, slopes, and x-intercepts. Mastering these concepts is essential for success in mathematics and related fields. Furthermore, this problem emphasizes the importance of critical thinking and problem-solving strategies, such as working backward and making logical deductions, which are valuable skills in any context. The ability to approach problems systematically and adapt to challenges, such as missing information, is a hallmark of effective problem solvers. Therefore, the lessons learned from this problem extend beyond the specific concepts of coordinate geometry and encompass broader problem-solving skills that are applicable to a wide range of situations.