Consider The Line $-x - 7y = 5$.1. Find The Equation Of The Line That Is Parallel To This Line And Passes Through The Point $(-2, 5$\]. Equation Of Parallel Line: $\square$2. Find The Equation Of The Line That Is

by ADMIN 215 views

Introduction

In mathematics, parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. The concept of parallel lines is crucial in geometry and is used to solve various problems in mathematics and physics. In this article, we will discuss 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 given point.

What are Parallel Lines?

Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. Two lines are parallel if they have the same slope and do not intersect at any point. The slope of a line is a measure of how steep it is and is calculated as the ratio of the vertical change (rise) to the horizontal change (run).

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. The slope-intercept form of a line is the most commonly used form and is used to find the equation of a line that passes through a given point.

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 given point, we need to follow these steps:

  1. Find the slope of the given line.
  2. Use the slope to find the equation of the parallel line.
  3. Substitute the coordinates of the given point into the equation of the parallel line to find the value of the y-intercept.

Step 1: Find the Slope of the Given Line

The given line is −x−7y=5-x - 7y = 5. To find the slope of this line, we need to rewrite it in the slope-intercept form. We can do this by solving for y:

y = (-1/7)x - (5/7)

The slope of the given line is -1/7.

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 -1/7. We can use this slope to find the equation of the parallel line. Let's call the equation of the parallel line y = mx + b. We know that m = -1/7, so we can substitute this value into the equation:

y = (-1/7)x + b

Step 3: Find the Value of the Y-Intercept

The parallel line passes through the point (-2, 5). We can substitute the coordinates of this point into the equation of the parallel line to find the value of the y-intercept:

5 = (-1/7)(-2) + b

Simplifying the equation, we get:

5 = 2/7 + b

Multiplying both sides of the equation by 7, we get:

35 = 2 + 7b

Subtracting 2 from both sides of the equation, we get:

33 = 7b

Dividing both sides of the equation by 7, we get:

b = 33/7

The value of the y-intercept is 33/7.

Equation of the Parallel Line

Now that we have found the value of the y-intercept, we can substitute it into the equation of the parallel line:

y = (-1/7)x + 33/7

Simplifying the equation, we get:

y = (-1/7)x + 33/7

Conclusion

In this article, we discussed 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 given point. We used the slope-intercept form of a line to find the equation of the parallel line and substituted the coordinates of the given point into the equation to find the value of the y-intercept. The equation of the parallel line is y = (-1/7)x + 33/7.

Example Problems

  1. Find the equation of the line that is parallel to the line 2x+3y=52x + 3y = 5 and passes through the point (1, 2).
  2. Find the equation of the line that is parallel to the line x−2y=3x - 2y = 3 and passes through the point (-1, 4).

Solutions

  1. The slope of the given line is -2/3. The equation of the parallel line is y = (-2/3)x + b. Substituting the coordinates of the given point into the equation, we get:

2 = (-2/3)(1) + b

Simplifying the equation, we get:

2 = -2/3 + b

Multiplying both sides of the equation by 3, we get:

6 = -2 + 3b

Adding 2 to both sides of the equation, we get:

8 = 3b

Dividing both sides of the equation by 3, we get:

b = 8/3

The equation of the parallel line is y = (-2/3)x + 8/3.

  1. The slope of the given line is 1/2. The equation of the parallel line is y = (1/2)x + b. Substituting the coordinates of the given point into the equation, we get:

4 = (1/2)(-1) + b

Simplifying the equation, we get:

4 = -1/2 + b

Multiplying both sides of the equation by 2, we get:

8 = -1 + 2b

Adding 1 to both sides of the equation, we get:

9 = 2b

Dividing both sides of the equation by 2, we get:

b = 9/2

The equation of the parallel line is y = (1/2)x + 9/2.

Final Thoughts

Introduction

In our previous article, we discussed 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 given point. In this article, we will answer some frequently asked questions about parallel lines and provide additional examples to help you understand the concept better.

Q&A

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, no matter how far they are extended. Perpendicular lines, on the other hand, are lines that intersect at a right angle (90 degrees).

Q: How do I know if two lines are parallel or perpendicular?

A: To determine if two lines are parallel or perpendicular, you can use the following methods:

  • Check if the lines have the same slope. If they do, they are parallel.
  • Check if the lines intersect at a right angle (90 degrees). If they do, they are perpendicular.
  • Use the slope-intercept form of a line to find the equation of each line. If the slopes are equal, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.

Q: Can two lines be both parallel and perpendicular at the same time?

A: No, two lines cannot be both parallel and perpendicular at the same time. If two lines are parallel, they will never intersect, and if two lines are perpendicular, they will intersect at a right angle.

Q: How do I find the equation of a line that is parallel to a given line and passes through a given point?

A: To find the equation of a line that is parallel to a given line and passes through a given point, follow these steps:

  1. Find the slope of the given line.
  2. Use the slope to find the equation of the parallel line.
  3. Substitute the coordinates of the given point into the equation of the parallel line to find the value of the y-intercept.

Q: What is the equation of a line that is parallel to the line 2x+3y=52x + 3y = 5 and passes through the point (1, 2)?

A: The slope of the given line is -2/3. The equation of the parallel line is y = (-2/3)x + b. Substituting the coordinates of the given point into the equation, we get:

2 = (-2/3)(1) + b

Simplifying the equation, we get:

2 = -2/3 + b

Multiplying both sides of the equation by 3, we get:

6 = -2 + 3b

Adding 2 to both sides of the equation, we get:

8 = 3b

Dividing both sides of the equation by 3, we get:

b = 8/3

The equation of the parallel line is y = (-2/3)x + 8/3.

Q: What is the equation of a line that is parallel to the line x−2y=3x - 2y = 3 and passes through the point (-1, 4)?

A: The slope of the given line is 1/2. The equation of the parallel line is y = (1/2)x + b. Substituting the coordinates of the given point into the equation, we get:

4 = (1/2)(-1) + b

Simplifying the equation, we get:

4 = -1/2 + b

Multiplying both sides of the equation by 2, we get:

8 = -1 + 2b

Adding 1 to both sides of the equation, we get:

9 = 2b

Dividing both sides of the equation by 2, we get:

b = 9/2

The equation of the parallel line is y = (1/2)x + 9/2.

Additional Examples

  1. Find the equation of the line that is parallel to the line 3x+2y=53x + 2y = 5 and passes through the point (2, 3).
  2. Find the equation of the line that is parallel to the line x+4y=3x + 4y = 3 and passes through the point (-2, 1).

Solutions

  1. The slope of the given line is -3/2. The equation of the parallel line is y = (-3/2)x + b. Substituting the coordinates of the given point into the equation, we get:

3 = (-3/2)(2) + b

Simplifying the equation, we get:

3 = -3 + b

Adding 3 to both sides of the equation, we get:

6 = b

The equation of the parallel line is y = (-3/2)x + 6.

  1. The slope of the given line is -1/4. The equation of the parallel line is y = (-1/4)x + b. Substituting the coordinates of the given point into the equation, we get:

1 = (-1/4)(-2) + b

Simplifying the equation, we get:

1 = 1/2 + b

Multiplying both sides of the equation by 2, we get:

2 = 1 + 2b

Subtracting 1 from both sides of the equation, we get:

1 = 2b

Dividing both sides of the equation by 2, we get:

b = 1/2

The equation of the parallel line is y = (-1/4)x + 1/2.

Conclusion

In this article, we answered some frequently asked questions about parallel lines and provided additional examples to help you understand the concept better. We hope that this article has been helpful in clarifying any doubts you may have had about parallel lines. If you have any further questions or need additional help, please don't hesitate to ask.