Write An Equation Whose Graph Is A Line Perpendicular To The Graph Of $x=-7$ And Which Passes Through The Point $(-7,1)$.
Introduction
In mathematics, the concept of perpendicular lines is a fundamental aspect of geometry and algebra. When dealing with lines, it's essential to understand how to identify and create equations for lines that are perpendicular to a given line. In this article, we will explore how to write an equation for a line that is perpendicular to the graph of $x=-7$ and passes through the point $(-7,1)$.
Understanding Perpendicular Lines
To begin, let's recall the definition of perpendicular lines. Two lines are perpendicular if they intersect at a right angle (90 degrees). In the context of the given problem, we are looking for a line that is perpendicular to the graph of $x=-7$. This means that the slope of the desired line will be the negative reciprocal of the slope of the given line.
The Equation of a Line
The equation of a line can be written in the form $y = mx + b$, where $m$ is the slope of the line and $b$ is the y-intercept. To find the equation of a line that passes through a given point, we can use the point-slope form of a line, which is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point.
Finding the Slope of the Given Line
The given line is $x=-7$, which is a vertical line. The slope of a vertical line is undefined, as it does not have a slope in the classical sense. However, we can still find the slope of a line that is perpendicular to this line. Since the slope of the given line is undefined, the slope of the perpendicular line will be 0.
Finding 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 $(-7,1)$. Plugging in the values of $x_1$, $y_1$, and $m$ into the point-slope form, we get:
Simplifying the equation, we get:
Adding 1 to both sides, we get:
This is the equation of the line that is perpendicular to the graph of $x=-7$ and passes through the point $(-7,1)$.
Conclusion
In this article, we have explored how to write an equation for a line that is perpendicular to the graph of $x=-7$ and passes through the point $(-7,1)$. We have used the concept of perpendicular lines and the equation of a line to find the desired equation. The final equation is $y = 1$, which represents a horizontal line that passes through the point $(-7,1)$.
Key Takeaways
- The slope of a vertical line is undefined.
- The slope of a line that is perpendicular to a vertical line is 0.
- The equation of a line can be written in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
- The point-slope form of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point.
Further Reading
For more information on perpendicular lines and the equation of a line, we recommend the following resources:
- Khan Academy: Perpendicular Lines
- Math Is Fun: Perpendicular Lines
- Wolfram MathWorld: Perpendicular Lines
By following the steps outlined in this article, you should be able to write an equation for a line that is perpendicular to the graph of $x=-7$ and passes through the point $(-7,1)$.
Introduction
In our previous article, we explored how to write an equation for a line that is perpendicular to the graph of $x=-7$ and passes through the point $(-7,1)$. In this article, we will answer some of the most frequently asked questions about perpendicular lines.
Q: What is the definition of perpendicular lines?
A: Perpendicular lines are two lines that intersect at a right angle (90 degrees). In other words, if two lines are perpendicular, they form a right angle when they intersect.
Q: How do I find the slope of a line that is perpendicular to a given line?
A: To find the slope of a line that is perpendicular to a given line, you need to find the negative reciprocal of the slope of the given line. If the slope of the given line is $m$, then the slope of the perpendicular line is $-\frac{1}{m}$.
Q: What is the equation of a line that is perpendicular to the graph of $x=-7$ and passes through the point $(-7,1)$?
A: The equation of a line that is perpendicular to the graph of $x=-7$ and passes through the point $(-7,1)$ is $y = 1$. This is because the slope of the perpendicular line is 0, and the point-slope form of a line is $y - y_1 = m(x - x_1)$.
Q: Can a line be perpendicular to itself?
A: No, a line cannot be perpendicular to itself. By definition, two lines are perpendicular if they intersect at a right angle. If a line intersects itself, it does not form a right angle, so it cannot be perpendicular to itself.
Q: How do I find the equation of a line that is perpendicular to a given line and passes through a given point?
A: To find the equation of a line that is perpendicular to a given line and passes through a given point, 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 of a line to find the equation of the perpendicular line.
Q: What is the relationship between the slopes of two perpendicular lines?
A: The slopes of two perpendicular lines are negative reciprocals of each other. In other words, if the slope of one line is $m$, then the slope of the perpendicular line is $-\frac{1}{m}$.
Q: Can a line be perpendicular to a horizontal line?
A: Yes, a line can be perpendicular to a horizontal line. In fact, a line that is perpendicular to a horizontal line is a vertical line.
Q: Can a line be perpendicular to a vertical line?
A: Yes, a line can be perpendicular to a vertical line. In fact, a line that is perpendicular to a vertical line is a horizontal line.
Conclusion
In this article, we have answered some of the most frequently asked questions about perpendicular lines. We hope that this article has provided you with a better understanding of perpendicular lines and how to find the equation of a line that is perpendicular to a given line and passes through a given point.
Key Takeaways
- Perpendicular lines are two lines that intersect at a right angle (90 degrees).
- The slope of a line that is perpendicular to a given line is the negative reciprocal of the slope of the given line.
- The equation of a line that is perpendicular to a given line and passes through a given point can be found using the point-slope form of a line.
- The slopes of two perpendicular lines are negative reciprocals of each other.
Further Reading
For more information on perpendicular lines, we recommend the following resources:
- Khan Academy: Perpendicular Lines
- Math Is Fun: Perpendicular Lines
- Wolfram MathWorld: Perpendicular Lines
By following the steps outlined in this article, you should be able to answer any questions about perpendicular lines and find the equation of a line that is perpendicular to a given line and passes through a given point.