Which Of The Following Is The Equation Of A Line That Passes Through $(4, -4)$ And Is Perpendicular To $y = 2x + 5$?A. \$y = 2x - 4$[/tex\]B. $y = 2x - 12$C. $y = -\frac{1}{2}x - 2$D.
Understanding the Problem
To find the equation of a line that passes through a given point and is perpendicular to another line, we need to understand the concept of slope and perpendicular lines. The slope of a line is a measure of how steep it is, and two lines are perpendicular if their slopes are negative reciprocals of each other.
Finding the Slope of the Given Line
The given line is in the form y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope of the given line is 2. To find the equation of a line that is perpendicular to this line, we need to find a line with a slope that is the negative reciprocal of 2.
Finding the Negative Reciprocal of the Slope
The negative reciprocal of 2 is -1/2. This means that the slope of the line we are looking for is -1/2.
Using the Point-Slope Form to Find the Equation of 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. We are given the point (4, -4) and the slope -1/2. Plugging these values into the point-slope form, we get:
y - (-4) = -1/2(x - 4)
Simplifying the Equation
To simplify the equation, we can start by distributing the -1/2 to the terms inside the parentheses:
y + 4 = -1/2x + 2
Rewriting the Equation in Slope-Intercept Form
To rewrite the equation in slope-intercept form, we can subtract 4 from both sides:
y = -1/2x - 2
Comparing the Equation to the Answer Choices
Now that we have found the equation of the line, we can compare it to the answer choices to see which one matches. The equation we found is y = -1/2x - 2, which matches answer choice C.
Conclusion
In conclusion, the equation of a line that passes through (4, -4) and is perpendicular to y = 2x + 5 is y = -1/2x - 2. This is answer choice C.
Step-by-Step Solution
Here is a step-by-step solution to the problem:
- Find the slope of the given line, which is 2.
- Find the negative reciprocal of the slope, which is -1/2.
- Use the point-slope form to find the equation of the line, plugging in the point (4, -4) and the slope -1/2.
- Simplify the equation by distributing the -1/2 to the terms inside the parentheses.
- Rewrite the equation in slope-intercept form by subtracting 4 from both sides.
- Compare the equation to the answer choices to see which one matches.
Key Concepts
- Slope: a measure of how steep a line is
- Perpendicular lines: two lines with slopes that are negative reciprocals of each other
- Point-slope form: a way to write the equation of a line using a point on the line and the slope
- Slope-intercept form: a way to write the equation of a line using the slope and the y-intercept
Common Mistakes
- Failing to find the negative reciprocal of the slope
- Failing to use the point-slope form to find the equation of the line
- Failing to simplify the equation
- Failing to rewrite the equation in slope-intercept form
Real-World Applications
- Finding the equation of a line that passes through a given point and is perpendicular to another line is an important concept in mathematics and has many real-world applications, such as:
- Designing buildings and bridges
- Creating computer graphics and animations
- Modeling population growth and disease spread
- Analyzing financial data and making predictions about the stock market
Practice Problems
- Find the equation of a line that passes through (2, 3) and is perpendicular to y = x - 2.
- Find the equation of a line that passes through (5, 1) and is perpendicular to y = 2x + 1.
- Find the equation of a line that passes through (1, 4) and is perpendicular to y = x + 3.
Conclusion
In conclusion, finding the equation of a line that passes through a given point and is perpendicular to another line is an important concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can find the equation of a line that meets these criteria.
Q: What is the slope of a line that is perpendicular to y = 2x + 5?
A: The slope of a line that is perpendicular to y = 2x + 5 is the negative reciprocal of 2, which is -1/2.
Q: How do I find the equation of a line that passes through a given point and is perpendicular to another line?
A: To find the equation of a line that passes through a given point and is perpendicular to another line, you need to follow these steps:
- Find the slope of the given line.
- Find the negative reciprocal of the slope.
- Use the point-slope form to find the equation of the line, plugging in the point and the slope.
- Simplify the equation by distributing the slope to the terms inside the parentheses.
- Rewrite the equation in slope-intercept form by subtracting the y-coordinate of the point from both sides.
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 negative reciprocal of a slope?
A: To find the negative reciprocal of a slope, you need to follow these steps:
- Take the reciprocal of the slope by flipping the fraction.
- Change the sign of the reciprocal to make it negative.
Q: What is the difference between the slope-intercept form and the point-slope 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. 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. The main difference between the two forms is that the slope-intercept form uses the y-intercept, while the point-slope form uses a point on the line.
Q: How do I know if two lines are perpendicular?
A: Two lines are perpendicular if their slopes are negative reciprocals of each other. To check if two lines are perpendicular, you need to find the slopes of both lines and see if they are negative reciprocals of each other.
Q: Can I use the point-slope form to find the equation of a line that passes through a given point and is parallel to another line?
A: No, you cannot use the point-slope form to find the equation of a line that passes through a given point and is parallel to another line. The point-slope form is used to find the equation of a line that passes through a given point and is perpendicular to another line.
Q: How do I find the equation of a line that passes through a given point and is parallel to another line?
A: To find the equation of a line that passes through a given point and is parallel to another line, you need to follow these steps:
- Find the slope of the given line.
- Use the point-slope form to find the equation of the line, plugging in the point and the slope.
- Simplify the equation by distributing the slope to the terms inside the parentheses.
- Rewrite the equation in slope-intercept form by subtracting the y-coordinate of the point from both sides.
Q: What is the difference between a line that is perpendicular to another line and a line that is parallel to another line?
A: A line that is perpendicular to another line has a slope that is the negative reciprocal of the slope of the other line. A line that is parallel to another line has the same slope as the other line.
Q: Can I use the slope-intercept form to find the equation of a line that passes through a given point and is perpendicular to another line?
A: Yes, you can use the slope-intercept form to find the equation of a line that passes through a given point and is perpendicular to another line. However, you need to find the slope of the line first and then use the point-slope form to find the equation of the line.
Q: How do I find the slope of a line that is perpendicular to another line?
A: To find the slope of a line that is perpendicular to another line, you need to find the negative reciprocal of the slope of the other line.
Q: What is the negative reciprocal of a slope?
A: The negative reciprocal of a slope is the reciprocal of the slope with a negative sign.
Q: Can I use the point-slope form to find the equation of a line that passes through a given point and is parallel to another line?
A: No, you cannot use the point-slope form to find the equation of a line that passes through a given point and is parallel to another line. The point-slope form is used to find the equation of a line that passes through a given point and is perpendicular to another line.
Q: How do I find the equation of a line that passes through a given point and is parallel to another line?
A: To find the equation of a line that passes through a given point and is parallel to another line, you need to follow these steps:
- Find the slope of the given line.
- Use the point-slope form to find the equation of the line, plugging in the point and the slope.
- Simplify the equation by distributing the slope to the terms inside the parentheses.
- Rewrite the equation in slope-intercept form by subtracting the y-coordinate of the point from both sides.
Q: What is the difference between a line that is perpendicular to another line and a line that is parallel to another line?
A: A line that is perpendicular to another line has a slope that is the negative reciprocal of the slope of the other line. A line that is parallel to another line has the same slope as the other line.
Q: Can I use the slope-intercept form to find the equation of a line that passes through a given point and is perpendicular to another line?
A: Yes, you can use the slope-intercept form to find the equation of a line that passes through a given point and is perpendicular to another line. However, you need to find the slope of the line first and then use the point-slope form to find the equation of the line.
Q: How do I find the slope of a line that is perpendicular to another line?
A: To find the slope of a line that is perpendicular to another line, you need to find the negative reciprocal of the slope of the other line.
Q: What is the negative reciprocal of a slope?
A: The negative reciprocal of a slope is the reciprocal of the slope with a negative sign.
Q: Can I use the point-slope form to find the equation of a line that passes through a given point and is parallel to another line?
A: No, you cannot use the point-slope form to find the equation of a line that passes through a given point and is parallel to another line. The point-slope form is used to find the equation of a line that passes through a given point and is perpendicular to another line.
Q: How do I find the equation of a line that passes through a given point and is parallel to another line?
A: To find the equation of a line that passes through a given point and is parallel to another line, you need to follow these steps:
- Find the slope of the given line.
- Use the point-slope form to find the equation of the line, plugging in the point and the slope.
- Simplify the equation by distributing the slope to the terms inside the parentheses.
- Rewrite the equation in slope-intercept form by subtracting the y-coordinate of the point from both sides.
Q: What is the difference between a line that is perpendicular to another line and a line that is parallel to another line?
A: A line that is perpendicular to another line has a slope that is the negative reciprocal of the slope of the other line. A line that is parallel to another line has the same slope as the other line.
Q: Can I use the slope-intercept form to find the equation of a line that passes through a given point and is perpendicular to another line?
A: Yes, you can use the slope-intercept form to find the equation of a line that passes through a given point and is perpendicular to another line. However, you need to find the slope of the line first and then use the point-slope form to find the equation of the line.
Q: How do I find the slope of a line that is perpendicular to another line?
A: To find the slope of a line that is perpendicular to another line, you need to find the negative reciprocal of the slope of the other line.
Q: What is the negative reciprocal of a slope?
A: The negative reciprocal of a slope is the reciprocal of the slope with a negative sign.
Q: Can I use the point-slope form to find the equation of a line that passes through a given point and is parallel to another line?
A: No, you cannot use the point-slope form to find the equation of a line that passes through a given point and is parallel to another line. The point-slope form is used to find the equation of a line that passes through a given point and is perpendicular to another line