Write The Equation Of A Line That Passes Through The Point { (0,1)$}$ And Is Perpendicular To { Y = 5x$}$.A) { Y = -\frac{1}{5}x + 1$}$B) { Y = 5x + 1$}$C) { Y = \frac{1}{5}x + 1$} D ) \[ D) \[ D ) \[ Y = X +
Introduction
In mathematics, linear equations are a fundamental concept that describe a relationship between two variables. When given a point and a line, we can find the equation of a line that passes through the point and is perpendicular to the given line. In this article, we will explore how to solve linear equations and find the equation of a line that meets these conditions.
Understanding Perpendicular Lines
Before we dive into the solution, let's understand what it means for two lines to be perpendicular. Two lines are perpendicular if they intersect at a right angle (90 degrees). In the context of linear equations, this means that the slopes of the two lines are negative reciprocals of each other.
The Given Line
The given line is {y = 5x$}$. This is a linear equation in slope-intercept form, where the slope (m) is 5 and the y-intercept (b) is 0.
Finding the Slope of the Perpendicular Line
To find the equation of a line that is perpendicular to the given line, we need to find the slope of the perpendicular line. Since the slopes of perpendicular lines are negative reciprocals of each other, we can find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
The slope of the given line is 5, so the slope of the perpendicular line is .
The Point-Slope Form
Now that we have the slope of the perpendicular line, we can use the point-slope form to find the equation of the line. The point-slope form is given by:
{y - y_1 = m(x - x_1)$}$
where {(x_1, y_1)$}$ is a point on the line and m is the slope.
In this case, we are given the point {(0, 1)$}$ and the slope . Plugging these values into the point-slope form, we get:
{y - 1 = -\frac{1}{5}(x - 0)$}$
Simplifying the Equation
To simplify the equation, we can distribute the slope to the terms inside the parentheses:
{y - 1 = -\frac{1}{5}x$}$
Next, we can add 1 to both sides of the equation to isolate y:
{y = -\frac{1}{5}x + 1$}$
Conclusion
In this article, we have shown how to find the equation of a line that passes through a given point and is perpendicular to a given line. We used the point-slope form and the concept of negative reciprocals to find the slope of the perpendicular line and then used the point-slope form to find the equation of the line.
The final answer is {y = -\frac{1}{5}x + 1$}$.
Discussion
- What is the relationship between the slopes of perpendicular lines?
- How do you find the slope of a line that is perpendicular to a given line?
- What is the point-slope form, and how is it used to find the equation of a line?
Answer Key
A) {y = -\frac{1}{5}x + 1$}$
Additional Resources
- Linear Equations
- Point-Slope Form
- Perpendicular Lines
Frequently Asked Questions: Linear Equations and Perpendicular Lines ====================================================================
Q: What is the relationship between the slopes of perpendicular lines?
A: The slopes of perpendicular lines are negative reciprocals of each other. This means that if the slope of one line is m, then the slope of the perpendicular line is -1/m.
Q: How do you 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 take the negative reciprocal of the slope of the given line. For example, if the slope of the given line is 5, then the slope of the perpendicular line is -1/5.
Q: What is the point-slope form, and how is it used to find the equation of a line?
A: The point-slope form is a way to write the equation of a line that passes through a given point and has a given slope. The point-slope form is given by:
{y - y_1 = m(x - x_1)$}$
where {(x_1, y_1)$}$ is a point on the line and m is the slope. To find the equation of a line using the point-slope form, you need to plug in the values of the point and the slope into the equation.
Q: How do you find the equation of a line that passes through a given point and is perpendicular to a given line?
A: To find the equation of a line that passes through a given point and is perpendicular to a given line, you need to follow these steps:
- Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
- Use the point-slope form to write the equation of the line.
- Simplify the equation by distributing the slope and combining like terms.
Q: What is the difference between the slope-intercept form and the point-slope form?
A: The slope-intercept form is a way to write the equation of a line that has a given slope and y-intercept. The slope-intercept form is given by:
{y = mx + b$}$
where m is the slope and b is the y-intercept. The point-slope form, on the other hand, is a way to write the equation of a line that passes through a given point and has a given slope.
Q: How do you determine if two lines are perpendicular?
A: To determine if two lines are perpendicular, you need to check if the slopes of the two lines are negative reciprocals of each other. If the slopes are negative reciprocals, then the lines are perpendicular.
Q: What is the significance of the y-intercept in the slope-intercept form?
A: The y-intercept in the slope-intercept form represents the point where the line intersects the y-axis. The y-intercept is the value of y when x is equal to 0.
Q: How do you find the equation of a line that passes through two given points?
A: To find the equation of a line that passes through two given points, you need to follow these steps:
- Find the slope of the line by using the formula:
{m = \frac{y_2 - y_1}{x_2 - x_1}$}$
where {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are the two points. 2. Use the point-slope form to write the equation of the line. 3. Simplify the equation by distributing the slope and combining like terms.
Q: What is the relationship between the x-intercept and the y-intercept in the slope-intercept form?
A: The x-intercept and the y-intercept in the slope-intercept form are related by the equation:
{x = -\frac{b}{m}$}$
where m is the slope and b is the y-intercept. The x-intercept is the value of x when y is equal to 0.
Q: How do you find the equation of a line that passes through a given point and has a given slope?
A: To find the equation of a line that passes through a given point and has a given slope, you need to follow these steps:
- Use the point-slope form to write the equation of the line.
- Simplify the equation by distributing the slope and combining like terms.
Q: What is the significance of the slope in the slope-intercept form?
A: The slope in the slope-intercept form represents the rate of change of the line. The slope is the value of the change in y divided by the change in x.
Q: How do you determine if a line is parallel to another line?
A: To determine if a line is parallel to another line, you need to check if the slopes of the two lines are equal. If the slopes are equal, then the lines are parallel.
Q: What is the relationship between the slope and the y-intercept in the slope-intercept form?
A: The slope and the y-intercept in the slope-intercept form are related by the equation:
{y = mx + b$}$
where m is the slope and b is the y-intercept. The slope represents the rate of change of the line, and the y-intercept represents the point where the line intersects the y-axis.
Q: How do you find the equation of a line that passes through a given point and is parallel to a given line?
A: To find the equation of a line that passes through a given point and is parallel to a given line, you need to follow these steps:
- Find the slope of the parallel line by using the slope of the given line.
- Use the point-slope form to write the equation of the line.
- Simplify the equation by distributing the slope and combining like terms.
Q: What is the significance of the point-slope form in finding the equation of a line?
A: The point-slope form is a useful tool for finding the equation of a line that passes through a given point and has a given slope. It allows you to write the equation of the line in a simple and concise way.
Q: How do you determine if a line is perpendicular to another line?
A: To determine if a line is perpendicular to another line, you need to check if the slopes of the two lines are negative reciprocals of each other. If the slopes are negative reciprocals, then the lines are perpendicular.