Consider The Line Y = 9 X − 4 Y = 9x - 4 Y = 9 X − 4 .1. Find The Equation Of The Line That Is Parallel To This Line And Passes Through The Point ( 7 , − 4 (7, -4 ( 7 , − 4 ]. Equation Of Parallel Line: $\square$2. Find The Equation Of The Line That Is
Introduction
In mathematics, parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. Parallel lines have the same slope, which is a measure of how steep the line is. In this article, we will explore the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point.
What are Parallel Lines?
Parallel lines are lines that have the same slope and never intersect. This means that if you draw two lines on a graph, and they never touch, they are parallel. Parallel lines can be vertical or horizontal, or they can be at an angle to each other.
The Equation of a Line
The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of points on the line. The general form of the equation of a line is:
y = mx + b
where m is the slope of the line and b is the y-intercept.
Finding the Equation of a Parallel Line
To find the equation of a line that is parallel to a given line and passes through a specific point, we need to follow these steps:
- Find the slope of the given line.
- Use the slope to find the equation of the parallel line.
- Substitute the coordinates of the point into the equation to find the value of b.
Step 1: Find the Slope of the Given Line
The given line is y = 9x - 4. To find the slope of this line, we can rewrite it in the form y = mx + b, where m is the slope.
y = 9x - 4
In this equation, the slope is 9.
Step 2: Find the Equation of the Parallel Line
Since the parallel line has the same slope as the given line, the slope of the parallel line is also 9. We can use this slope to find the equation of the parallel line.
y = 9x + b
Step 3: Substitute the Coordinates of the Point into the Equation
The point (7, -4) lies on the parallel line. We can substitute the coordinates of this point into the equation to find the value of b.
-4 = 9(7) + b
-4 = 63 + b
b = -67
Now that we have found the value of b, we can write the equation of the parallel line.
y = 9x - 67
Equation of Parallel Line
The equation of the parallel line is y = 9x - 67.
Conclusion
In this article, we have explored the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point. We have followed the steps to find the slope of the given line, use the slope to find the equation of the parallel line, and substitute the coordinates of the point into the equation to find the value of b. The equation of the parallel line is y = 9x - 67.
Example Problems
- Find the equation of the line that is parallel to the line y = 2x + 5 and passes through the point (3, 11).
- Find the equation of the line that is parallel to the line y = -3x + 2 and passes through the point (4, -5).
Solutions
- The slope of the given line is 2. The equation of the parallel line is y = 2x + b. Substituting the coordinates of the point (3, 11) into the equation, we get:
11 = 2(3) + b
11 = 6 + b
b = 5
The equation of the parallel line is y = 2x + 5.
- The slope of the given line is -3. The equation of the parallel line is y = -3x + b. Substituting the coordinates of the point (4, -5) into the equation, we get:
-5 = -3(4) + b
-5 = -12 + b
b = 7
The equation of the parallel line is y = -3x + 7.
Applications of Parallel Lines
Parallel lines have many applications in mathematics and real-life situations. Some examples include:
- Geometry: Parallel lines are used to define the properties of angles and shapes.
- Trigonometry: Parallel lines are used to solve problems involving right triangles and trigonometric functions.
- Physics: Parallel lines are used to describe the motion of objects and the forces acting on them.
- Engineering: Parallel lines are used to design and build structures such as bridges and buildings.
Conclusion
Q: What is the difference between parallel lines and perpendicular lines?
A: Parallel lines are lines that lie in the same plane and never intersect, while perpendicular lines are lines that intersect at a 90-degree angle.
Q: How do you find the equation of a line that is parallel to a given line and passes through a specific point?
A: To find the equation of a line that is parallel to a given line and passes through a specific point, you need to follow these steps:
- Find the slope of the given line.
- Use the slope to find the equation of the parallel line.
- Substitute the coordinates of the point into the equation to find the value of b.
Q: What is the slope of a line that is parallel to the line y = 2x + 5?
A: The slope of the line y = 2x + 5 is 2. Therefore, the slope of any line that is parallel to this line is also 2.
Q: How do you find the equation of a line that is parallel to the line y = 2x + 5 and passes through the point (3, 11)?
A: To find the equation of a line that is parallel to the line y = 2x + 5 and passes through the point (3, 11), you need to follow these steps:
- Find the slope of the given line, which is 2.
- Use the slope to find the equation of the parallel line, which is y = 2x + b.
- Substitute the coordinates of the point (3, 11) into the equation to find the value of b.
11 = 2(3) + b
11 = 6 + b
b = 5
The equation of the parallel line is y = 2x + 5.
Q: What is the equation of a line that is parallel to the line y = -3x + 2 and passes through the point (4, -5)?
A: To find the equation of a line that is parallel to the line y = -3x + 2 and passes through the point (4, -5), you need to follow these steps:
- Find the slope of the given line, which is -3.
- Use the slope to find the equation of the parallel line, which is y = -3x + b.
- Substitute the coordinates of the point (4, -5) into the equation to find the value of b.
-5 = -3(4) + b
-5 = -12 + b
b = 7
The equation of the parallel line is y = -3x + 7.
Q: How do you find the equation of a line that is parallel to a given line and passes through two specific points?
A: To find the equation of a line that is parallel to a given line and passes through two specific points, you need to follow these steps:
- Find the slope of the given line.
- Use the slope to find the equation of the parallel line.
- Substitute the coordinates of one of the points into the equation to find the value of b.
- Substitute the coordinates of the other point into the equation to find the value of b.
Q: What is the equation of a line that is parallel to the line y = 2x + 5 and passes through the points (3, 11) and (4, 13)?
A: To find the equation of a line that is parallel to the line y = 2x + 5 and passes through the points (3, 11) and (4, 13), you need to follow these steps:
- Find the slope of the given line, which is 2.
- Use the slope to find the equation of the parallel line, which is y = 2x + b.
- Substitute the coordinates of one of the points, (3, 11), into the equation to find the value of b.
11 = 2(3) + b
11 = 6 + b
b = 5
The equation of the parallel line is y = 2x + 5.
Q: How do you find the equation of a line that is parallel to a given line and passes through three specific points?
A: To find the equation of a line that is parallel to a given line and passes through three specific points, you need to follow these steps:
- Find the slope of the given line.
- Use the slope to find the equation of the parallel line.
- Substitute the coordinates of two of the points into the equation to find the value of b.
- Substitute the coordinates of the third point into the equation to find the value of b.
Q: What is the equation of a line that is parallel to the line y = 2x + 5 and passes through the points (3, 11), (4, 13), and (5, 15)?
A: To find the equation of a line that is parallel to the line y = 2x + 5 and passes through the points (3, 11), (4, 13), and (5, 15), you need to follow these steps:
- Find the slope of the given line, which is 2.
- Use the slope to find the equation of the parallel line, which is y = 2x + b.
- Substitute the coordinates of two of the points, (3, 11) and (4, 13), into the equation to find the value of b.
11 = 2(3) + b
11 = 6 + b
b = 5
The equation of the parallel line is y = 2x + 5.
Conclusion
In conclusion, parallel lines are an important concept in mathematics that have many applications in real-life situations. By understanding the concept of parallel lines and how to find the equation of a line that is parallel to a given line and passes through a specific point, we can solve problems involving geometry, trigonometry, physics, and engineering.