Equation Of A Parallel Line Through A Point A Step By Step Guide

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In the realm of coordinate geometry, a fundamental concept involves determining the equation of a line that adheres to specific conditions. One common scenario is finding the equation of a line that is parallel to a given line and passes through a particular point. This exploration will delve into the step-by-step process of tackling such problems, providing a clear and comprehensive guide for anyone seeking to master this skill. We'll start with a detailed explanation of the underlying principles, followed by a practical application to the specific problem at hand.

Understanding Parallel Lines and Their Equations

Before diving into the problem, it's crucial to grasp the concept of parallel lines and how their equations are represented in slope-intercept form. Parallel lines, by definition, are lines that never intersect. This geometric property translates directly into their algebraic representation: parallel lines have the same slope. The slope-intercept form of a linear equation, y = mx + b, is particularly useful here, where m represents the slope and b represents the y-intercept.

Knowing that parallel lines share the same slope is the cornerstone of solving this type of problem. When given the equation of a line, you can immediately identify its slope, which then becomes the slope of any line parallel to it. The challenge then shifts to finding the specific y-intercept that makes the parallel line pass through the given point. This is where the point-slope form or substitution methods come into play.

To illustrate further, consider two parallel lines. If one line has a slope of 2, any line parallel to it will also have a slope of 2. The only difference between their equations will be the y-intercept. One line might intersect the y-axis at 3, while the other might intersect at -1. However, their steepness and direction remain identical, ensuring they never meet. This fundamental understanding of the relationship between slope and parallelism is essential for solving the equation we are about to tackle. Mastering this concept not only helps in solving specific problems but also builds a solid foundation for more advanced topics in coordinate geometry and calculus.

Step-by-Step Solution to Find the Parallel Line

To find the equation of the line parallel to y = (6/5)x + 10 and passing through the point (12, -2), we will follow a structured approach:

  1. Identify the Slope of the Given Line: The equation y = (6/5)x + 10 is in slope-intercept form, making it easy to see that the slope, m, is 6/5. Since parallel lines have the same slope, the line we are looking for will also have a slope of 6/5.
  2. Use the Point-Slope Form or Slope-Intercept Form: We now know the slope of the parallel line and a point it passes through. We can use either the point-slope form or directly substitute into the slope-intercept form to find the equation.
    • Point-Slope Form: The point-slope form of a linear equation is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Substituting the point (12, -2) and the slope 6/5, we get: y - (-2) = (6/5)(x - 12). This equation can then be simplified to the slope-intercept form.
    • Slope-Intercept Form: Alternatively, we can use the slope-intercept form, y = mx + b. We know m = 6/5, so the equation becomes y = (6/5)x + b. To find b, we substitute the coordinates of the point (12, -2) into the equation: -2 = (6/5)(12) + b. Solving for b will give us the y-intercept.
  3. Solve for the y-intercept (b): If we used the slope-intercept form, we now have an equation with b as the only unknown. Let's solve for b:
    • -2 = (6/5)(12) + b
    • -2 = 72/5 + b
    • b = -2 - 72/5
    • b = -10/5 - 72/5
    • b = -82/5
  4. Write the Equation of the Parallel Line: Now that we have the slope m = 6/5 and the y-intercept b = -82/5, we can write the equation of the parallel line in slope-intercept form: y = (6/5)x - 82/5. This equation represents the unique line that is parallel to the given line and passes through the specified point.

By following these steps, anyone can confidently find the equation of a line parallel to a given line and passing through a specific point. This process involves understanding the fundamental properties of parallel lines, applying the appropriate formulas, and carefully solving for the unknowns.

Applying the Solution to the Given Problem

Now, let's apply this method to the specific problem posed: Finding the equation of a line parallel to y = (6/5)x + 10 and passing through the point (12, -2).

  1. Identify the Slope: As discussed, the slope of the given line y = (6/5)x + 10 is 6/5. Therefore, the parallel line will also have a slope of 6/5.
  2. Use the Point-Slope Form: Substituting the point (12, -2) and the slope 6/5 into the point-slope form, we get:
    • y - (-2) = (6/5)(x - 12)
    • y + 2 = (6/5)(x - 12)
  3. Simplify to Slope-Intercept Form: To make the equation easier to compare with the provided options, we'll convert it to slope-intercept form:
    • y + 2 = (6/5)x - (6/5)(12)
    • y + 2 = (6/5)x - 72/5
    • y = (6/5)x - 72/5 - 2
    • y = (6/5)x - 72/5 - 10/5
    • y = (6/5)x - 82/5

Therefore, the equation of the line parallel to y = (6/5)x + 10 and passing through the point (12, -2) is y = (6/5)x - 82/5. This result can be compared with the given options to identify the correct answer. This step-by-step application underscores the practical use of the method and its ability to solve real-world problems involving parallel lines.

Analyzing the Answer Choices

Having derived the equation y = (6/5)x - 82/5, let's analyze the provided answer choices to identify the correct one. This involves comparing our derived equation with each option, focusing on both the slope and the y-intercept. This critical step ensures accuracy and reinforces the understanding of linear equations.

We are looking for an equation with a slope of 6/5 and a y-intercept of -82/5. Examining the options:

  • y = (6/5)x + 10: This line has the correct slope of 6/5 but an incorrect y-intercept of 10. Thus, it is not the correct answer.
  • y = (6/5)x + 12: Similar to the first option, this line has the correct slope but a different y-intercept of 12. Therefore, it's not the solution we seek.
  • y = -(5/6)x - 10: This line has a slope of -5/6, which is the negative reciprocal of 6/5. This indicates that the line is perpendicular, not parallel, to the given line. Consequently, this option is incorrect.
  • y = (5/6)x - 12: This line has a slope of 5/6, which is different from the required slope of 6/5. Hence, this option is also incorrect.

After careful comparison, we notice that none of the provided answer choices exactly match our derived equation, y = (6/5)x - 82/5. This discrepancy highlights the importance of thorough calculation and the possibility of errors in provided options. In a real-world scenario, this would prompt a re-evaluation of the problem and the answer choices to ensure accuracy.

The Importance of Double-Checking

This situation emphasizes the significance of double-checking calculations and comparing results with given options. It also showcases that sometimes, mistakes can occur in the provided information, making it crucial to have a strong understanding of the underlying principles to identify and rectify such errors. In this case, the correct equation, y = (6/5)x - 82/5, was not among the choices, prompting a need for further scrutiny or clarification.

Conclusion: Mastering Parallel Line Equations

In conclusion, finding the equation of a line parallel to a given line and passing through a specific point involves a clear, step-by-step process. Understanding the properties of parallel lines, particularly their equal slopes, is paramount. The point-slope form and slope-intercept form are valuable tools in this endeavor, allowing us to construct the equation of the desired line.

This exploration has not only demonstrated the method but also highlighted the importance of careful calculation, comparison with answer choices, and the awareness that errors can occur in provided information. Mastering these skills is essential for success in coordinate geometry and related mathematical fields. The ability to confidently tackle such problems provides a strong foundation for more advanced concepts and real-world applications.

By practicing these techniques and understanding the underlying principles, anyone can confidently approach problems involving parallel lines and linear equations, solidifying their mathematical prowess and problem-solving abilities. This mastery extends beyond the classroom, equipping individuals with the analytical skills necessary for various fields and disciplines that rely on geometric and algebraic reasoning.