Find The Line Perpendicular To $y = -3x + 7$ That Includes The Point $(-3, 1)$.The Equation Of The Line Is: $y - 1 = \frac{1}{3}(x + 3)$.Remember: $ Y − Y 1 = M ( X − X 1 ) Y - Y_1 = M(x - X_1) Y − Y 1 = M ( X − X 1 ) [/tex].
Introduction
In mathematics, finding the equation of a line that is perpendicular to a given line and passes through a specific point is a fundamental concept. This problem involves finding the equation of a line that is perpendicular to the line $y = -3x + 7$ and includes the point $(-3, 1)$. In this article, we will walk you through the step-by-step process of finding the equation of the perpendicular line.
Understanding the Problem
To find the equation of the perpendicular line, we need to recall 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 is calculated as the ratio of the vertical change (rise) to the horizontal change (run). The point-slope form of a line is given by the equation $y - y_1 = m(x - x_1)$, where $m$ is the slope of the line, and $(x_1, y_1)$ is a point on the line.
Finding the Slope of the Given Line
The given line is $y = -3x + 7$. To find the slope of this line, we can rewrite it in the slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this case, the slope of the given line is $-3$.
Finding the Slope of the Perpendicular Line
Since the perpendicular line is perpendicular to the given line, its slope is the negative reciprocal of the slope of the given line. The negative reciprocal of $-3$ is $\frac{1}{3}$.
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 the point $(-3, 1)$, which we can use as $(x_1, y_1)$. Plugging in the values, we get:
Simplifying the Equation
To simplify the equation, we can multiply both sides by 3 to eliminate the fraction:
Expanding the left-hand side, we get:
Adding 3 to both sides, we get:
Conclusion
In this article, we found the equation of the line that is perpendicular to the line $y = -3x + 7$ and includes the point $(-3, 1)$. The equation of the perpendicular line is $y - 1 = \frac{1}{3}(x + 3)$. We used the concept of slope and the point-slope form of a line to find the equation of the perpendicular line. This problem demonstrates the importance of understanding the concept of slope and the point-slope form of a line in mathematics.
Key Takeaways
- The slope of a line is a measure of how steep it is.
- The point-slope form of a line is given by the equation $y - y_1 = m(x - x_1)$.
- The slope of the perpendicular line is the negative reciprocal of the slope of the given line.
- To find the equation of the perpendicular line, we can use the point-slope form and plug in the values of the slope and the point.
Real-World Applications
Finding the equation of a line that is perpendicular to a given line and passes through a specific point has many real-world applications. For example, in engineering, it is used to design buildings and bridges. In physics, it is used to describe the motion of objects. In computer science, it is used in algorithms and data structures.
Common Mistakes
When finding the equation of a line that is perpendicular to a given line and passes through a specific point, there are several common mistakes to avoid. These include:
- Not using the correct slope of the given line.
- Not using the correct point on the line.
- Not simplifying the equation correctly.
- Not checking the work for errors.
Conclusion
In conclusion, finding the equation of a line that is perpendicular to a given line and passes through a specific point is a fundamental concept in mathematics. It involves understanding the concept of slope and the point-slope form of a line. By following the steps outlined in this article, you can find the equation of the perpendicular line and apply it to real-world problems.
Introduction
In our previous article, we discussed how to find the equation of a line that is perpendicular to a given line and passes through a specific point. In this article, we will answer some of the most frequently asked questions about finding the perpendicular line.
Q: What is the slope of the perpendicular line?
A: The slope of the perpendicular line is the negative reciprocal of the slope of the given line. For example, if the slope of the given line is -3, the slope of the perpendicular line is 1/3.
Q: How do I find the equation of the perpendicular line?
A: To find the equation of the perpendicular line, you can use the point-slope form of a line, which is given by the equation y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line.
Q: What is the point-slope form of a line?
A: The point-slope form of a line is given by the equation y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line.
Q: How do I simplify the equation of the perpendicular line?
A: To simplify the equation of the perpendicular line, you can multiply both sides of the equation by the denominator to eliminate the fraction. You can also add or subtract the same value to both sides of the equation to isolate the variable.
Q: What are some common mistakes to avoid when finding the equation of the perpendicular line?
A: Some common mistakes to avoid when finding the equation of the perpendicular line include:
- Not using the correct slope of the given line.
- Not using the correct point on the line.
- Not simplifying the equation correctly.
- Not checking the work for errors.
Q: How do I check my work for errors?
A: To check your work for errors, you can plug in the values of the slope and the point into the equation of the perpendicular line and see if it is true. You can also graph the line and check if it is perpendicular to the given line.
Q: What are some real-world applications of finding the equation of the perpendicular line?
A: Finding the equation of the perpendicular line has many real-world applications, including:
- Designing buildings and bridges in engineering.
- Describing the motion of objects in physics.
- Using algorithms and data structures in computer science.
Q: Can I use the equation of the perpendicular line to find the equation of a line that is parallel to the given line?
A: No, the equation of the perpendicular line is used to find the equation of a line that is perpendicular to the given line, not parallel. To find the equation of a line that is parallel to the given line, you can use the slope of the given line and the point-slope form of a line.
Q: Can I use the equation of the perpendicular line to find the equation of a line that passes through a specific point and has a given slope?
A: Yes, you can use the equation of the perpendicular line to find the equation of a line that passes through a specific point and has a given slope. Simply plug in the values of the slope and the point into the equation of the perpendicular line and solve for the variable.
Conclusion
In conclusion, finding the equation of a line that is perpendicular to a given line and passes through a specific point is a fundamental concept in mathematics. By understanding the concept of slope and the point-slope form of a line, you can find the equation of the perpendicular line and apply it to real-world problems. We hope that this Q&A guide has been helpful in answering some of the most frequently asked questions about finding the perpendicular line.