Finding The Equation Of A Perpendicular Line A Step-by-Step Guide
In mathematics, particularly in coordinate geometry, determining the equation of a line that is perpendicular to another line and passes through a specific point is a fundamental concept. This problem combines the understanding of slopes, perpendicularity, and the point-slope form of a linear equation. In this comprehensive guide, we will delve into the process of finding the equation of such a line, using a step-by-step approach and clear explanations.
Understanding the Problem
The problem presented asks us to identify the equation of a line that satisfies two crucial conditions: it must be perpendicular to a given line, and it must pass through a specific point. The given line is represented by the equation y = (3/2)x + 1, and the point through which the perpendicular line must pass is (-12, 6). To solve this problem, we need to understand the relationship between the slopes of perpendicular lines and utilize the point-slope form of a linear equation.
Step 1: Determine the Slope of the Given Line
The given line is in slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. From the equation y = (3/2)x + 1, we can directly identify the slope as 3/2. This value is crucial because the slope of a line perpendicular to this one will have a specific relationship to it.
Step 2: Calculate the Slope of the Perpendicular Line
The key concept here is that perpendicular lines have slopes that are negative reciprocals of each other. This means that if a line has a slope of m, a line perpendicular to it will have a slope of -1/m. In our case, the slope of the given line is 3/2. Therefore, the slope of the line perpendicular to it will be the negative reciprocal of 3/2, which is -2/3. Understanding this negative reciprocal relationship is essential for solving this type of problem.
Step 3: Use the Point-Slope Form to Find the Equation
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 of the perpendicular line is -2/3, and we are given the point (-12, 6) through which the line passes. Plugging these values into the point-slope form, we get:
y - 6 = (-2/3)(x - (-12))
Simplifying this equation will give us the equation of the perpendicular line.
Step 4: Simplify the Equation to Slope-Intercept Form
To simplify the equation, we first distribute the -2/3 on the right side:
y - 6 = (-2/3)x - 8
Next, we add 6 to both sides to isolate y:
y = (-2/3)x - 2
This equation is now in slope-intercept form (y = mx + b), where we can clearly see the slope (-2/3) and the y-intercept (-2).
Step 5: Verify the Solution
To ensure our solution is correct, we can verify that the line y = (-2/3)x - 2 indeed passes through the point (-12, 6). Substitute x = -12 into the equation:
y = (-2/3)(-12) - 2
y = 8 - 2
y = 6
Since the y-value is 6 when x is -12, the point (-12, 6) lies on the line, confirming our solution. Furthermore, the slope of -2/3 is indeed the negative reciprocal of the original slope 3/2, confirming the perpendicularity.
The Correct Answer
Based on our step-by-step solution, the equation of the line that is perpendicular to y = (3/2)x + 1 and passes through (-12, 6) is y = (-2/3)x - 2. This corresponds to option B in the given choices.
Analyzing Incorrect Options
Understanding why the other options are incorrect is just as important as understanding why the correct answer is correct. This helps to solidify your understanding of the concepts and avoid making similar mistakes in the future.
Option A: y = (-2/3)x - 16
This option has the correct slope (-2/3), which indicates a perpendicular line. However, when we substitute the point (-12, 6) into this equation, we get:
6 = (-2/3)(-12) - 16
6 = 8 - 16
6 = -8
This is not true, so the line does not pass through the point (-12, 6). The error here is likely in the calculation of the y-intercept.
Option C: y = (3/2)x - 21
This option has the same slope as the original line (3/2), which means the lines are parallel, not perpendicular. Therefore, this option is incorrect. It is a common mistake to forget to take the negative reciprocal when finding the slope of a perpendicular line.
Key Concepts and Takeaways
- Perpendicular Lines: 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.
- Slope-Intercept Form: The equation y = mx + b represents a line in slope-intercept form, where m is the slope and b is the y-intercept.
- Point-Slope Form: The equation y - y1 = m( x - x1) represents a line in point-slope form, where (x1, y1) is a point on the line and m is the slope.
- Substituting Points: To check if a point lies on a line, substitute the x and y coordinates of the point into the equation of the line. If the equation holds true, the point lies on the line.
Additional Practice Problems
To further solidify your understanding of this concept, try solving the following practice problems:
- Find the equation of the line perpendicular to y = -2x + 3 and passing through the point (4, -1).
- Find the equation of the line perpendicular to y = (1/4)x - 5 and passing through the point (-8, 2).
- Find the equation of the line perpendicular to y = 5x + 1 and passing through the point (0, 7).
By working through these problems, you will gain confidence in your ability to find the equations of perpendicular lines.
Conclusion
Finding the equation of a line perpendicular to a given line and passing through a specific point is a fundamental skill in coordinate geometry. By understanding the relationship between slopes of perpendicular lines and utilizing the point-slope form, you can solve these problems effectively. Remember to always verify your solution to ensure accuracy. This comprehensive guide has provided you with the necessary steps, explanations, and practice problems to master this concept. Keep practicing, and you'll become proficient in solving these types of problems.
In summary, the correct answer to the original problem is B. y = (-2/3)x - 2. This equation represents the line that is perpendicular to y = (3/2)x + 1 and passes through the point (-12, 6).
