Finding Equations Of Parallel Lines A Step-by-Step Guide

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This article will guide you through the process of finding the equation of a line that meets specific criteria. We'll focus on a common problem in coordinate geometry: determining the equation of a line that passes through a given point and is parallel to another given line. This concept is fundamental in various mathematical and real-world applications, from determining trajectories to understanding rates of change. To fully grasp this, we will delve into the properties of parallel lines and how their slopes relate, allowing you to confidently tackle such problems. Understanding parallel lines and their equations is crucial not only for academic success in mathematics but also for various practical applications in fields such as engineering, physics, and computer graphics.

Key Concepts: Parallel Lines and Slope-Intercept Form

Before diving into the solution, let's refresh two crucial concepts:

  • Parallel Lines: Parallel lines are lines in the same plane that never intersect. A fundamental property of parallel lines is that they have the same slope. The slope of a line indicates its steepness and direction. If two lines have the same slope, they increase or decrease at the same rate, ensuring they never meet.
  • Slope-Intercept Form: The slope-intercept form of a linear equation is y = mx + b, where m represents the slope of the line, and b represents the y-intercept (the point where the line crosses the y-axis). This form is incredibly useful because it explicitly shows the slope and y-intercept, making it easy to graph the line and compare it to other lines.

Understanding these concepts is essential for solving the problem at hand. The slope-intercept form provides a clear framework for representing linear equations, and the property of parallel lines having equal slopes is the key to finding the desired equation.

Step-by-Step Solution

Now, let's tackle the problem step by step. Our goal is to find the equation of a line that passes through the point (6, 14) and is parallel to the line y = -4/3x - 1. Here’s how we can do it:

  1. Identify the Slope: The given equation is in slope-intercept form (y = mx + b). By comparing y = -4/3x - 1 with the general form, we can see that the slope of the given line is -4/3. Remember, the coefficient of x in the slope-intercept form represents the slope. Since we are looking for a line parallel to this one, the line we want to find will have the same slope, -4/3. This is the fundamental property of parallel lines – they have equal slopes. This identification is the first critical step because it gives us the m value for our new equation.
  2. Use the 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. We know the slope (m = -4/3) and a point that the line passes through (6, 14). Plug these values into the point-slope form. Substituting the known values into the point-slope form allows us to create an equation specific to the line we are trying to find. Our equation becomes y - 14 = -4/3(x - 6). This form is particularly useful when you have a point and a slope, as it allows you to directly create an equation representing the line.
  3. Convert to Slope-Intercept Form: While the point-slope form is a valid equation for the line, it’s often useful to convert it to slope-intercept form (y = mx + b) for easier interpretation and comparison. To do this, we'll simplify the equation we obtained in the previous step. Converting to slope-intercept form allows us to clearly see the y-intercept of the line, which can be useful for graphing and further analysis. First, distribute the -4/3 on the right side: y - 14 = -4/3x + 8. Then, add 14 to both sides to isolate y: y = -4/3x + 8 + 14, which simplifies to y = -4/3x + 22. This is the equation of the line in slope-intercept form.

By following these steps, we have successfully found the equation of the line that passes through the point (6, 14) and is parallel to the line y = -4/3x - 1. The key was to understand the properties of parallel lines, use the point-slope form effectively, and then convert the equation to slope-intercept form for clarity.

Verifying the Solution

To ensure our answer is correct, we can verify it in a couple of ways:

  • Check the Slope: We found the equation y = -4/3x + 22. The slope of this line is -4/3, which is the same as the slope of the given line y = -4/3x - 1. This confirms that the lines are indeed parallel. This verification step is crucial to ensure we haven't made any errors in our calculations. The slopes matching is a fundamental requirement for parallel lines.
  • Check the Point: To make sure the line passes through the point (6, 14), substitute x = 6 into the equation y = -4/3x + 22 and see if we get y = 14. Substituting the given point into the equation is a direct way to confirm that the line passes through that point. Calculation: y = -4/3(6) + 22 = -8 + 22 = 14. This confirms that the point (6, 14) lies on the line.

Since both the slope and the point conditions are satisfied, we can be confident that our solution is correct. This verification process reinforces the importance of checking our work to avoid errors and build confidence in our problem-solving abilities.

The Answer

Therefore, the equation of the line that passes through the point (6, 14) and is parallel to the line y = -4/3x - 1 is:

A. y = -4/3x + 22

Why Other Options Are Incorrect

Understanding why the other options are incorrect is as important as understanding why the correct answer is correct. This helps in reinforcing the concepts and avoiding common mistakes.

  • B. y = 3/4x + 20: This line has a slope of 3/4. This slope is the negative reciprocal of -4/3, which means this line is perpendicular, not parallel, to the given line. Parallel lines must have the same slope, not a negative reciprocal slope.
  • C. y = 3/4x + 8: This line also has a slope of 3/4, making it perpendicular to the given line, not parallel. The different y-intercept simply shifts the line up or down, but the slope determines its direction. Additionally, if you substitute the point (6, 14) into this equation, you get y = 3/4(6) + 8 = 4.5 + 8 = 12.5, which is not equal to 14, meaning the line does not pass through the point (6, 14).

By analyzing why these options are incorrect, we strengthen our understanding of parallel lines and the conditions necessary for a line to pass through a given point. This critical analysis is a valuable skill in problem-solving.

Conclusion

In conclusion, finding the equation of a line that is parallel to another line and passes through a specific point involves understanding the properties of parallel lines and the effective use of slope-intercept and point-slope forms. We identified the slope of the given line, used the point-slope form to create an equation for the parallel line, and converted it to slope-intercept form for clarity. We also verified our solution and discussed why the other options were incorrect.

This process not only helps in solving this specific problem but also builds a strong foundation for tackling more complex problems in coordinate geometry. Remember, the key concepts are:

  • Parallel lines have the same slope.
  • The slope-intercept form (y = mx + b) is useful for identifying slope and y-intercept.
  • The point-slope form (y - y1 = m(x - x1)) is useful when you have a point and a slope.

By mastering these concepts and practicing problem-solving techniques, you can confidently approach similar questions and excel in your understanding of linear equations.

Practice Problems

To further solidify your understanding, try these practice problems:

  1. Find the equation of the line that passes through the point (-2, 5) and is parallel to the line y = 2x - 3.
  2. What is the equation of the line that passes through the point (1, -1) and is parallel to the line y = -x + 4?
  3. Determine the equation of the line that passes through the point (0, 3) and is parallel to the line y = 1/2x + 1.

Working through these problems will help you reinforce the steps and concepts discussed in this article, allowing you to confidently tackle a variety of problems involving parallel lines.

By mastering these concepts and practicing problem-solving techniques, you can confidently approach similar questions and excel in your understanding of linear equations. Remember, mathematics is not just about finding the right answer, but about understanding the process and reasoning behind it. This deeper understanding will serve you well in your academic journey and beyond.