Identify An Equation In Point-slope Form For The Line Perpendicular To $y=\frac{1}{4} X-7$ That Passes Through $(-2,-6$\].A. $y+6=-\frac{1}{4}(x+2$\] B. $y+2=-4(x+6$\] C. $y+6=-4(x+2$\] D.
Introduction
In mathematics, the point-slope form of a linear equation is a powerful tool for describing lines in a coordinate plane. Given a line with a known slope and a point through which it passes, we can use the point-slope form to write an equation for the line. In this article, we will explore how to identify an equation in point-slope form for a line that is perpendicular to a given line and passes through a specific point.
Understanding the Point-Slope Form
The point-slope form of a linear equation is given by the formula:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line. This formula allows us to write an equation for a line that passes through a specific point and has a known slope.
Finding the Slope of the Given Line
The given line is described by the equation y = (1/4)x - 7. To find the slope of this line, we can rewrite the equation in the slope-intercept form, which is given by the formula:
y = mx + b
where m is the slope and b is the y-intercept. Comparing this formula to the given equation, we can see that the slope of the given line is 1/4.
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 1/4 is -4.
Writing the Equation of the Perpendicular Line
We are given that the perpendicular line passes through the point (-2, -6). Using the point-slope form, we can write an equation for the line as follows:
y - (-6) = -4(x - (-2))
Simplifying this equation, we get:
y + 6 = -4(x + 2)
Comparing the Options
Now that we have written the equation of the perpendicular line, we can compare it to the options given in the problem. The equation we derived is:
y + 6 = -4(x + 2)
This equation matches option C.
Conclusion
In this article, we have explored how to identify an equation in point-slope form for a line that is perpendicular to a given line and passes through a specific point. We used the point-slope form of a linear equation and the concept of negative reciprocals to find the slope of the perpendicular line and write its equation. We then compared our derived equation to the options given in the problem and found that it matches option C.
Key Takeaways
- The point-slope form of a linear equation is a powerful tool for describing lines in a coordinate plane.
- The slope of a line is the negative reciprocal of the slope of a line that is perpendicular to it.
- The point-slope form can be used to write an equation for a line that passes through a specific point and has a known slope.
Additional Examples
- Find the equation of a line that is perpendicular to the line y = 2x + 3 and passes through the point (1, 2).
- Find the equation of a line that is perpendicular to the line y = -x + 2 and passes through the point (-1, 1).
Solutions
- The equation of the line that is perpendicular to the line y = 2x + 3 and passes through the point (1, 2) is y - 2 = -1/2(x - 1).
- The equation of the line that is perpendicular to the line y = -x + 2 and passes through the point (-1, 1) is y - 1 = 1(x + 1).
References
- [1] "Point-Slope Form of a Linear Equation." Math Open Reference, mathopenref.com/calgebra-point-slope.html.
- [2] "Slope of a Line." Math Is Fun, mathisfun.com/algebra/slope.html.
Frequently Asked Questions: Identifying Equations in Point-Slope Form ====================================================================
Q: What is the point-slope form of a linear equation?
A: The point-slope form of a linear equation is a formula that allows us to write an equation for a line that passes through a specific point and has a known slope. The formula is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line.
Q: How do I find the slope of a line in point-slope form?
A: To find the slope of a line in point-slope form, you need to rewrite the equation in the slope-intercept form, which is given by the formula:
y = mx + b
where m is the slope and b is the y-intercept. Once you have rewritten the equation in slope-intercept form, you can identify the slope (m) as the coefficient of x.
Q: What is the negative reciprocal of a slope?
A: The negative reciprocal of a slope is a value that is equal to the negative of the reciprocal of the slope. For example, if the slope is 2, the negative reciprocal is -1/2. The negative reciprocal of a slope is used to find the slope of a line that is perpendicular to the original line.
Q: How do I find the equation of a line that is perpendicular to a given line?
A: To find the equation of a line that is perpendicular to a given line, you need to follow these steps:
- Find the slope of the given line.
- Find the negative reciprocal of the slope of the given line.
- Use the point-slope form to write an equation for the line that is perpendicular to the given line.
Q: What is the point-slope form of a line that passes through a specific point and has a known slope?
A: The point-slope form of a line that passes through a specific point and has a known slope is given by the formula:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line.
Q: How do I simplify a point-slope form equation?
A: To simplify a point-slope form equation, you need to follow these steps:
- Distribute the slope (m) to the terms inside the parentheses.
- Combine like terms.
- Rewrite the equation in a more simplified form.
Q: What are some common mistakes to avoid when working with point-slope form equations?
A: Some common mistakes to avoid when working with point-slope form equations include:
- Forgetting to distribute the slope (m) to the terms inside the parentheses.
- Not combining like terms.
- Not rewriting the equation in a more simplified form.
Q: How do I use point-slope form equations in real-world applications?
A: Point-slope form equations are used in a variety of real-world applications, including:
- Finding the equation of a line that passes through a specific point and has a known slope.
- Finding the equation of a line that is perpendicular to a given line.
- Modeling real-world situations, such as the motion of an object or the growth of a population.
Q: What are some additional resources for learning about point-slope form equations?
A: Some additional resources for learning about point-slope form equations include:
- Online tutorials and videos.
- Math textbooks and workbooks.
- Online math communities and forums.
Conclusion
In this article, we have answered some frequently asked questions about identifying equations in point-slope form. We have covered topics such as the point-slope form of a linear equation, finding the slope of a line, and simplifying point-slope form equations. We have also discussed some common mistakes to avoid and provided some additional resources for learning about point-slope form equations.