Given That Another Line \[$ L_3 \$\] Is Parallel To Line \[$ L \$\], And Passes Through The Point \[$ (1, 2) \$\] Where The Equation Of Line \[$ L \$\] Is \[$ Y = 3x + 9 \$\]:(i) Find The Equation Of \[$ L_3
Introduction
In geometry, parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. Given a line and a point not on the line, we can find the equation of a parallel line that passes through the given point. In this article, we will discuss how to find the equation of a parallel line that passes through a given point, using the equation of the original line and the coordinates of the point.
Understanding the Problem
We are given a line { l $}$ with the equation { y = 3x + 9 $}$. We are also given a point { (1, 2) $}$ that lies on the line { L_3 $}$, which is parallel to the line { l $}$. Our goal is to find the equation of the line { L_3 $}$.
Recalling the Properties of Parallel Lines
Two lines are parallel if and only if they have the same slope. 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) between two points on the line. In the case of the line { l $}$, the slope is 3, which means that for every 1 unit we move to the right, we move up 3 units.
Finding the Slope of the Parallel Line
Since the line { L_3 $}$ is parallel to the line { l $}$, it must have the same slope as the line { l $}$. Therefore, the slope of the line { L_3 $}$ is also 3.
Using the Point-Slope Form
The point-slope form of a line is given by the equation { y - y_1 = m(x - x_1) $}$, where { (x_1, y_1) $}$ is a point on the line and { m $}$ is the slope of the line. We can use this form to find the equation of the line { L_3 $}$.
Substituting the Values
We know that the slope of the line { L_3 $}$ is 3, and the point { (1, 2) $}$ lies on the line. We can substitute these values into the point-slope form to get:
{ y - 2 = 3(x - 1) $}$
Simplifying the Equation
We can simplify the equation by distributing the 3 to the terms inside the parentheses:
{ y - 2 = 3x - 3 $}$
Adding 2 to Both Sides
To isolate the y-term, we can add 2 to both sides of the equation:
{ y = 3x - 1 $}$
Conclusion
In this article, we discussed how to find the equation of a parallel line that passes through a given point. We used the equation of the original line and the coordinates of the point to find the equation of the parallel line. We also recalled the properties of parallel lines and used the point-slope form to find the equation of the line. The final equation of the line { L_3 $}$ is { y = 3x - 1 $}$.
Example Problems
- Find the equation of a line that passes through the point { (2, 3) $}$ and is parallel to the line { y = 2x + 1 $}$.
- Find the equation of a line that passes through the point { (1, 4) $}$ and is parallel to the line { y = x - 2 $}$.
Solutions
- The slope of the line { y = 2x + 1 $}$ is 2, so the slope of the parallel line is also 2. We can use the point-slope form to find the equation of the line:
{ y - 3 = 2(x - 2) $}$
Simplifying the equation, we get:
{ y - 3 = 2x - 4 $}$
Adding 3 to both sides, we get:
{ y = 2x - 1 $}$
- The slope of the line { y = x - 2 $}$ is 1, so the slope of the parallel line is also 1. We can use the point-slope form to find the equation of the line:
{ y - 4 = 1(x - 1) $}$
Simplifying the equation, we get:
{ y - 4 = x - 1 $}$
Adding 4 to both sides, we get:
{ y = x + 3 $}$
Final Answer
Introduction
In our previous article, we discussed how to find the equation of a parallel line that passes through a given point. We used the equation of the original line and the coordinates of the point to find the equation of the parallel line. In this article, we will answer some frequently asked questions about finding the equation of a parallel line.
Q: What is the slope of a parallel line?
A: The slope of a parallel line is the same as the slope of the original line. This is because parallel lines have the same steepness, and the slope is a measure of how steep a line is.
Q: How do I find the equation of a parallel line that passes through a given point?
A: To find the equation of a parallel line that passes through a given point, you can use the point-slope form of a line. The point-slope form is given by the equation { y - y_1 = m(x - x_1) $}$, where { (x_1, y_1) $}$ is a point on the line and { m $}$ is the slope of the line.
Q: What if I don't know the slope of the original line?
A: If you don't know the slope of the original line, you can find it by using the equation of the line and the coordinates of a point on the line. For example, if the equation of the line is { y = mx + b $}$ and the coordinates of a point on the line are { (x_1, y_1) $}$, you can substitute these values into the equation to find the slope:
{ y_1 = mx_1 + b $}$
Solving for { m $}$, you get:
{ m = \frac{y_1 - b}{x_1} $}$
Q: Can I find the equation of a parallel line if I only know the equation of the original line and the coordinates of a point on the original line?
A: Yes, you can find the equation of a parallel line if you only know the equation of the original line and the coordinates of a point on the original line. To do this, you can use the point-slope form of a line and the fact that the slope of the parallel line is the same as the slope of the original line.
Q: How do I know if two lines are parallel?
A: Two lines are parallel if they have the same slope and lie in the same plane. You can check if two lines are parallel by finding their slopes and comparing them. If the slopes are the same, the lines are parallel.
Q: Can I find the equation of a parallel line if I only know the coordinates of two points on the original line?
A: Yes, you can find the equation of a parallel line if you only know the coordinates of two points on the original line. To do this, you can use the two-point form of a line, which is given by the equation { y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) $}$, where { (x_1, y_1) $}$ and { (x_2, y_2) $}$ are the coordinates of the two points.
Q: What if I want to find the equation of a parallel line that passes through a given point and is perpendicular to the original line?
A: To find the equation of a parallel line that passes through a given point and is perpendicular to the original line, you can use the point-slope form of a line and the fact that the slope of the perpendicular line is the negative reciprocal of the slope of the original line.
Conclusion
In this article, we answered some frequently asked questions about finding the equation of a parallel line. We discussed how to find the slope of a parallel line, how to find the equation of a parallel line that passes through a given point, and how to check if two lines are parallel. We also discussed how to find the equation of a parallel line that passes through a given point and is perpendicular to the original line.