Perpendicular Line Equation: Find The Solution!

by SLV Team 48 views

Hey guys! Let's dive into the fascinating world of linear equations and tackle a common question: Which equation represents a line perpendicular to a given line? In this case, we're looking at the line y = (3/8)x - 1. Understanding perpendicular lines is crucial in geometry and algebra, and we're here to break it down step by step.

Understanding Slope and Perpendicular Lines

Before we jump into the options, let's quickly review the concept of slope and how it relates to perpendicular lines. The slope of a line tells us how steep it is and in what direction it's going. A line in the form y = mx + b has a slope of m. So, for our given line, y = (3/8)x - 1, the slope is 3/8. Remember this value, as it's super important for finding our perpendicular line.

Now, what about perpendicular lines? Well, two lines are perpendicular if they intersect at a right angle (90 degrees). The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other. This means you flip the fraction and change the sign. So, if a line has a slope of m, a line perpendicular to it will have a slope of -1/m. For instance, if a line has a slope of 2, a perpendicular line has a slope of -1/2. This negative reciprocal relationship is the key to solving our problem.

In our case, the original line has a slope of 3/8. To find the slope of a line perpendicular to it, we flip the fraction and change the sign. So, the slope of the perpendicular line will be -8/3. Now we know that any equation representing a line perpendicular to y = (3/8)x - 1 must have a slope of -8/3 when written in slope-intercept form (y = mx + b). We can now use this information to evaluate the answer choices and pinpoint the correct equation. Remember, we're looking for an equation that, when rearranged into the y = mx + b format, gives us a slope of -8/3. This is our guiding principle as we examine the provided options.

Analyzing the Answer Choices

Let's examine the answer choices provided, one by one, to determine which one represents a line perpendicular to y = (3/8)x - 1:

  • A. 3x + 8y = -40 To determine the slope, we need to rearrange this equation into slope-intercept form (y = mx + b). Let's isolate y: 8y = -3x - 40. Now, divide both sides by 8: y = (-3/8)x - 5. The slope of this line is -3/8. This is the reciprocal of 3/8 but not the negative reciprocal, so it's not perpendicular.
  • B. 3x - 8y = -56 Again, let's rearrange to slope-intercept form: -8y = -3x - 56. Divide both sides by -8: y = (3/8)x + 7. The slope of this line is 3/8, which is the same as the original line's slope. This means the lines are parallel, not perpendicular. This option is definitely out.
  • C. 3y - 8x = 3 Let's get y by itself: 3y = 8x + 3. Divide both sides by 3: y = (8/3)x + 1. The slope here is 8/3. This is the reciprocal of 3/8, but it's not the negative reciprocal. So, while it's a step in the right direction, it's not quite what we're looking for.
  • D. 8x + 3y = 9 Rearranging for y: 3y = -8x + 9. Divide both sides by 3: y = (-8/3)x + 3. The slope of this line is -8/3. Bingo! This is the negative reciprocal of the original slope (3/8), so this line is perpendicular.

Therefore, after carefully analyzing each option and applying our understanding of perpendicular lines and their slopes, we've determined that the correct answer is D. 8x + 3y = 9. Guys, this illustrates the importance of understanding the relationship between slopes of perpendicular lines and how to manipulate equations into slope-intercept form.

Why Option D is the Correct Answer

Option D, 8x + 3y = 9, is the correct answer because, as we demonstrated, its slope is the negative reciprocal of the original line's slope. When we rearranged the equation into slope-intercept form (y = mx + b), we found the equation y = (-8/3)x + 3. The slope, -8/3, perfectly matches the requirement for a perpendicular line. This option showcases the practical application of the negative reciprocal concept. It's crucial to remember that for two lines to be perpendicular, the product of their slopes must be -1. In this case, (3/8) * (-8/3) = -1, confirming their perpendicularity. This key insight solidifies our understanding of why option D is the solution.

Furthermore, let's visualize these lines. The original line, y = (3/8)x - 1, has a positive slope, meaning it rises from left to right. The perpendicular line, y = (-8/3)x + 3, has a negative slope, meaning it falls from left to right. The steepness of the perpendicular line is greater than the original line, reflecting the steeper slope (-8/3 compared to 3/8). Imagining these lines intersecting at a right angle helps to further cement the concept of perpendicularity in our minds. Guys, this visual understanding is super beneficial for tackling similar problems in the future.

Key Takeaways and Tips

Let's recap the key takeaways from this problem so you're well-equipped to tackle similar questions in the future:

  1. Perpendicular Lines: Two lines are perpendicular if they intersect at a right angle (90 degrees).
  2. Negative Reciprocal Slopes: The slopes of perpendicular lines are negative reciprocals of each other. If one line has a slope of m, a line perpendicular to it has a slope of -1/m.
  3. Slope-Intercept Form: The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. This form is crucial for identifying the slope of a line.
  4. Rearranging Equations: You need to be comfortable rearranging equations into slope-intercept form to easily identify the slope.
  5. Visualizing Lines: Try to visualize the lines and their slopes. This can help you understand the relationship between perpendicular lines.

Here are some tips for solving problems involving perpendicular lines:

  • Identify the Slope: First, determine the slope of the given line. If it's not in slope-intercept form, rearrange it.
  • Find the Negative Reciprocal: Calculate the negative reciprocal of the slope you found in the previous step. This will be the slope of any line perpendicular to the given line.
  • Check the Answer Choices: Examine the answer choices and see which equation, when rearranged into slope-intercept form, has the slope you calculated in step 2.
  • Double-Check: If you have time, double-check your answer by ensuring that the product of the slopes of the two lines is -1.

By mastering these concepts and following these tips, you'll be a pro at solving perpendicular line problems. Remember guys, practice makes perfect, so keep working through examples to solidify your understanding!

Practice Problems

Now that we've thoroughly discussed the concept and solution, let's test your understanding with a couple of practice problems:

  1. Which equation represents a line perpendicular to y = -2x + 5?
    • A. y = 2x - 3
    • B. y = -2x + 1
    • C. y = (1/2)x + 4
    • D. y = (-1/2)x - 2
  2. What is the equation of a line perpendicular to 4x - y = 7 and passing through the point (0, 2)?
    • A. y = 4x + 2
    • B. y = (-1/4)x + 2
    • C. y = (-4)x + 2
    • D. y = (1/4)x + 2

Work through these problems using the strategies we discussed. Pay close attention to finding the negative reciprocal slope and rearranging equations into slope-intercept form. Don't be afraid to visualize the lines to help you understand the relationships. Solving these practice problems will significantly boost your confidence in tackling similar questions.

Conclusion

In conclusion, guys, understanding the relationship between slopes of perpendicular lines is a fundamental concept in algebra and geometry. By knowing that perpendicular lines have slopes that are negative reciprocals of each other, you can confidently solve problems like the one we tackled today. Remember to practice rearranging equations into slope-intercept form and visualizing the lines to deepen your understanding. With consistent effort and a clear grasp of the principles, you'll be able to conquer any perpendicular line challenge that comes your way. Keep up the great work, and remember that mathematics is all about building a strong foundation of knowledge and skills. You've got this! Hitting these concepts head-on will set you up for success in more advanced math topics too!