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

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In the realm of linear equations, understanding the concept of parallel lines is crucial. Parallel lines, by definition, never intersect, maintaining a constant distance from each other. This property translates into a specific mathematical relationship: parallel lines have the same slope. This article dives deep into how to determine the equation of a line that is parallel to a given line and passes through a specific point. We will use the principles of slope-intercept form and point-slope form to solve a sample problem. This comprehensive guide will help you master the concepts behind finding equations of parallel lines, complete with detailed explanations and step-by-step solutions. By the end of this article, you will have a solid grasp of how to approach similar problems confidently and accurately.

Understanding Slope and Parallel Lines

Before we dive into solving the problem, it’s essential to understand the concept of slope and its relationship to parallel lines. The slope of a line is a measure of its steepness and direction. It represents the change in the y-coordinate for every unit change in the x-coordinate. In the slope-intercept form of a linear equation, which is y = mx + b, the coefficient m represents the slope, and b represents the y-intercept (the point where the line crosses the y-axis).

Parallel lines have the same slope. This is a fundamental principle in coordinate geometry. If two lines have the same slope, they will never intersect, regardless of their y-intercepts. Conversely, if two lines have different slopes, they will intersect at some point. To find a line parallel to a given line, the first and most critical step is to identify the slope of the given line. This slope will be the same for the parallel line we are trying to find. Understanding this core concept is crucial for tackling problems involving parallel lines.

The Given Equation and the Parallel Slope

Our problem presents us with the equation y = (5/2)x + 31. This equation is in slope-intercept form, making it straightforward to identify the slope. By comparing the equation to the standard form y = mx + b, we can see that the slope m is 5/2. Therefore, any line parallel to this line will also have a slope of 5/2. This realization is the first key step in solving the problem. We now know that the equation of the line we are looking for will have the form y = (5/2)x + b, where b is the y-intercept that we need to determine. The challenge now shifts to finding the correct y-intercept that will make the line pass through the given point.

Using the Point-Slope Form

To find the equation of a line that passes through a specific point, we can use the point-slope form of a linear equation. The point-slope form is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful when you know a point on the line and the slope, which is precisely the information we have in this problem. We know the slope of the parallel line is 5/2, and we know it passes through the point (-10, 30). Plugging these values into the point-slope form, we get:

y - 30 = (5/2)(x - (-10)) This equation represents the line we are looking for, but it is not yet in the slope-intercept form. The next step is to simplify this equation and convert it to the slope-intercept form to match the answer choices provided.

Converting to Slope-Intercept Form

Now, we will simplify the point-slope equation and convert it to slope-intercept form (y = mx + b). Starting with the equation y - 30 = (5/2)(x + 10), we first distribute the 5/2 across the terms inside the parentheses: y - 30 = (5/2)x + (5/2)(10) This simplifies to: y - 30 = (5/2)x + 25 Next, we isolate y by adding 30 to both sides of the equation: y = (5/2)x + 25 + 30 This gives us the final equation in slope-intercept form: y = (5/2)x + 55 This equation represents the line that is parallel to y = (5/2)x + 31 and passes through the point (-10, 30). Comparing this result with the given options, we find that option D matches our solution. This step-by-step conversion is critical in ensuring we arrive at the correct equation.

Analyzing the Answer Choices

To ensure we have the correct answer, it’s helpful to analyze the answer choices provided. The options are:

A. y = (-2/5)x + 2 B. y = (-2/5)x + 26 C. y = (5/2)x - 85 D. y = (5/2)x + 55

We are looking for a line that is parallel to y = (5/2)x + 31, which means it must have the same slope of 5/2. Options A and B have a slope of -2/5, which is the negative reciprocal of 5/2. This means these lines are perpendicular, not parallel, to the given line. Therefore, options A and B can be immediately eliminated. Option C has the correct slope of 5/2 but a different y-intercept. While it is parallel, it does not pass through the point (-10, 30). Only option D has the correct slope of 5/2 and the correct y-intercept, making it the correct answer. This process of elimination is a useful strategy in multiple-choice questions.

The Correct Answer and Conclusion

After carefully analyzing the problem and working through the steps, we have determined that the correct answer is D. y = (5/2)x + 55. This equation represents the line that is parallel to y = (5/2)x + 31 and passes through the point (-10, 30). Understanding the concepts of slope, parallel lines, and the point-slope form is crucial for solving problems like this. By following a systematic approach, you can confidently tackle similar problems. Remember to first identify the slope, use the point-slope form to create an equation, and then convert to slope-intercept form to find the final answer. This structured methodology will help you navigate through linear equation problems effectively.

To recap, here’s a step-by-step solution to the problem:

  1. Identify the slope of the given line: In the equation y = (5/2)x + 31, the slope is 5/2.
  2. Parallel lines have the same slope: The line parallel to the given line will also have a slope of 5/2.
  3. Use the point-slope form: The point-slope form is y - y₁ = m(x - x₁). Plug in the point (-10, 30) and the slope 5/2: y - 30 = (5/2)(x - (-10))
  4. Simplify the equation: y - 30 = (5/2)(x + 10)
  5. Distribute the slope: y - 30 = (5/2)x + 25
  6. Isolate y: y = (5/2)x + 25 + 30
  7. Final equation: y = (5/2)x + 55

When solving problems involving parallel lines, there are several common mistakes to avoid:

  • Incorrectly identifying the slope: Always ensure you correctly identify the slope from the given equation. Forgetting to consider the sign or misreading the coefficient can lead to incorrect answers.
  • Using the negative reciprocal for parallel lines: Remember, parallel lines have the same slope, not the negative reciprocal. The negative reciprocal is used for perpendicular lines.
  • Making algebraic errors: Be careful when distributing and simplifying equations. A small error can lead to a wrong answer.
  • Forgetting to convert to slope-intercept form: If the answer choices are in slope-intercept form, make sure to convert your equation to that form before comparing.
  • Not checking the point: Ensure that the final equation actually passes through the given point by plugging in the x and y coordinates.

To further solidify your understanding, try solving these practice problems:

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

In conclusion, finding the equation of a line parallel to a given line involves understanding the concept of slope and using the point-slope form effectively. By identifying the slope, applying the point-slope form, and simplifying the equation, you can confidently solve these types of problems. Remember to avoid common mistakes and practice regularly to enhance your skills. This guide provides a comprehensive approach to mastering parallel lines and linear equations, equipping you with the knowledge to tackle a wide range of mathematical challenges. Mastering these fundamental skills in linear equations opens the door to more advanced topics in mathematics. Keep practicing, and you'll become proficient in solving these types of problems!