What Is The Equation Of The Line That Passes Through The Point { (2, -2)$}$ And Is Perpendicular To The Line { X - 5y = 20$}$?
Introduction
In mathematics, finding the equation of a line that passes through a given point and is perpendicular to another line is a common problem. This problem involves using the concept of slope and the point-slope form of a line to find the equation of the desired line. In this article, we will discuss how to find the equation of the line that passes through the point (2, -2) and is perpendicular to the line x - 5y = 20.
Understanding the Problem
To solve this problem, we need to understand the concept of slope and the point-slope form of a line. The slope of a line is a measure of how steep it is, and it can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are two points on the line.
The point-slope form of a line is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Finding the Slope of the Given Line
The given line is x - 5y = 20. To find the slope of this line, we need to rewrite it in the slope-intercept form, which is:
y = mx + b
where m is the slope and b is the y-intercept.
To rewrite the given line in the slope-intercept form, we need to isolate y. We can do this by subtracting x from both sides of the equation and then dividing both sides by -5.
x - 5y = 20 -5y = -x + 20 y = (1/5)x - 4
Now that we have the slope-intercept form of the given line, we can see that the slope is 1/5.
Finding the Slope of the Perpendicular Line
Since the line we are looking for is perpendicular to the given line, its slope will be the negative reciprocal of the slope of the given line. The negative reciprocal of 1/5 is -5.
Using the Point-Slope Form to Find the Equation of the Perpendicular Line
Now that we have the slope of the perpendicular line, we can use the point-slope form to find its equation. We are given that the line passes through the point (2, -2), so we can substitute this point into the point-slope form along with the slope.
y - (-2) = -5(x - 2) y + 2 = -5x + 10 y = -5x + 8
Conclusion
In this article, we discussed how to find the equation of the line that passes through the point (2, -2) and is perpendicular to the line x - 5y = 20. We used the concept of slope and the point-slope form of a line to find the equation of the desired line. The final equation of the line is y = -5x + 8.
Additional Information
- The slope of the given line is 1/5.
- The slope of the perpendicular line is -5.
- The point-slope form of a line is given by y - y1 = m(x - x1).
- The slope-intercept form of a line is given by y = mx + b.
Example Problems
- Find the equation of the line that passes through the point (3, 1) and is perpendicular to the line 2x + 3y = 5.
- Find the equation of the line that passes through the point (4, -3) and is perpendicular to the line x - 2y = 7.
Step-by-Step Solutions
- To find the equation of the line that passes through the point (3, 1) and is perpendicular to the line 2x + 3y = 5, we need to follow these steps:
- Find the slope of the given line.
- Find the slope of the perpendicular line.
- Use the point-slope form to find the equation of the perpendicular line.
- To find the equation of the line that passes through the point (4, -3) and is perpendicular to the line x - 2y = 7, we need to follow these steps:
- Find the slope of the given line.
- Find the slope of the perpendicular line.
- Use the point-slope form to find the equation of the perpendicular line.
Common Mistakes
- One common mistake is to confuse the slope of the given line with the slope of the perpendicular line.
- Another common mistake is to forget to use the point-slope form to find the equation of the perpendicular line.
Tips and Tricks
- To find the equation of the line that passes through a given point and is perpendicular to another line, we need to use the concept of slope and the point-slope form of a line.
- We can use the slope-intercept form to find the slope of the given line.
- The negative reciprocal of the slope of the given line is the slope of the perpendicular line.
Conclusion
In this article, we discussed how to find the equation of the line that passes through the point (2, -2) and is perpendicular to the line x - 5y = 20. We used the concept of slope and the point-slope form of a line to find the equation of the desired line. The final equation of the line is y = -5x + 8.
Introduction
In our previous article, we discussed how to find the equation of a line that passes through a given point and is perpendicular to another line. In this article, we will answer some frequently asked questions related to this topic.
Q: What is the slope of the line that passes through the point (2, -2) and is perpendicular to the line x - 5y = 20?
A: The slope of the given line is 1/5. Since the line we are looking for is perpendicular to the given line, its slope will be the negative reciprocal of the slope of the given line. The negative reciprocal of 1/5 is -5.
Q: How do I find the equation of the line that passes through the point (3, 1) and is perpendicular to the line 2x + 3y = 5?
A: To find the equation of the line that passes through the point (3, 1) and is perpendicular to the line 2x + 3y = 5, you need to follow these steps:
- Find the slope of the given line.
- Find the slope of the perpendicular line.
- Use the point-slope form to find the equation of the perpendicular line.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Q: How do I find the slope of the given line?
A: To find the slope of the given line, you need to rewrite it in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.
Q: What is the slope-intercept form of a line?
A: The slope-intercept form of a line is given by y = mx + b, where m is the slope and b is the y-intercept.
Q: How do I find the equation of the line that passes through the point (4, -3) and is perpendicular to the line x - 2y = 7?
A: To find the equation of the line that passes through the point (4, -3) and is perpendicular to the line x - 2y = 7, you need to follow these steps:
- Find the slope of the given line.
- Find the slope of the perpendicular line.
- Use the point-slope form to find the equation of the perpendicular line.
Q: What is the negative reciprocal of the slope of the given line?
A: The negative reciprocal of the slope of the given line is the slope of the perpendicular line.
Q: How do I use the point-slope form to find the equation of the perpendicular line?
A: To use the point-slope form to find the equation of the perpendicular line, you need to substitute the point and the slope into the point-slope form.
Q: What is the final equation of the line that passes through the point (2, -2) and is perpendicular to the line x - 5y = 20?
A: The final equation of the line that passes through the point (2, -2) and is perpendicular to the line x - 5y = 20 is y = -5x + 8.
Conclusion
In this article, we answered some frequently asked questions related to finding the equation of a line that passes through a given point and is perpendicular to another line. We hope that this article has been helpful in clarifying any doubts you may have had on this topic.
Additional Information
- The slope of the given line is 1/5.
- The slope of the perpendicular line is -5.
- The point-slope form of a line is given by y - y1 = m(x - x1).
- The slope-intercept form of a line is given by y = mx + b.
- The negative reciprocal of the slope of the given line is the slope of the perpendicular line.
Example Problems
- Find the equation of the line that passes through the point (3, 1) and is perpendicular to the line 2x + 3y = 5.
- Find the equation of the line that passes through the point (4, -3) and is perpendicular to the line x - 2y = 7.
Step-by-Step Solutions
- To find the equation of the line that passes through the point (3, 1) and is perpendicular to the line 2x + 3y = 5, you need to follow these steps:
- Find the slope of the given line.
- Find the slope of the perpendicular line.
- Use the point-slope form to find the equation of the perpendicular line.
- To find the equation of the line that passes through the point (4, -3) and is perpendicular to the line x - 2y = 7, you need to follow these steps:
- Find the slope of the given line.
- Find the slope of the perpendicular line.
- Use the point-slope form to find the equation of the perpendicular line.
Common Mistakes
- One common mistake is to confuse the slope of the given line with the slope of the perpendicular line.
- Another common mistake is to forget to use the point-slope form to find the equation of the perpendicular line.
Tips and Tricks
- To find the equation of the line that passes through a given point and is perpendicular to another line, you need to use the concept of slope and the point-slope form of a line.
- You can use the slope-intercept form to find the slope of the given line.
- The negative reciprocal of the slope of the given line is the slope of the perpendicular line.