Determine The Relationship Of The Following Lines.${ \begin{array}{l} y = 3x - 4 \ y = \frac{1}{3}x + 4 \end{array} }$Select The Correct Response:A. The Lines Are NOT Parallel Because The Slopes Are Not Negative Reciprocals. B. The Lines
Introduction
In mathematics, particularly in algebra and geometry, understanding the relationship between two lines is crucial for solving various problems. Two lines can be parallel, perpendicular, or neither. In this article, we will determine the relationship between two given lines and explore the conditions for parallel and perpendicular lines.
What are Parallel Lines?
Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. They have the same slope but different y-intercepts. 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).
What are Perpendicular Lines?
Perpendicular lines are lines that intersect at a right angle (90 degrees). They have slopes that are negative reciprocals of each other. The negative reciprocal of a slope is obtained by flipping the fraction and changing the sign of the numerator and denominator.
The Given Lines
We are given two lines:
Determining the Relationship
To determine the relationship between the two lines, we need to compare their slopes. The slope of the first line is 3, and the slope of the second line is .
Are the Lines Parallel?
To check if the lines are parallel, we need to see if their slopes are negative reciprocals of each other. The negative reciprocal of 3 is , which is not equal to . Therefore, the lines are not parallel.
Are the Lines Perpendicular?
To check if the lines are perpendicular, we need to see if their slopes are negative reciprocals of each other. The negative reciprocal of is , which is not equal to 3. Therefore, the lines are not perpendicular.
Conclusion
Based on the analysis, we can conclude that the two lines are neither parallel nor perpendicular. They intersect at a point, and their slopes are not negative reciprocals of each other.
Final Answer
The correct response is:
A. The lines are NOT parallel because the slopes are not negative reciprocals.
Additional Information
To find the point of intersection, we can set the two equations equal to each other and solve for x.
Solving for x, we get:
Substituting x into one of the original equations, we get:
Therefore, the point of intersection is (3, 5).
Summary
Introduction
In our previous article, we determined the relationship between two given lines and explored the conditions for parallel and perpendicular lines. In this article, we will answer some frequently asked questions related to the topic.
Q: What is the difference between parallel and perpendicular lines?
A: Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. They have the same slope but different y-intercepts. Perpendicular lines, on the other hand, are lines that intersect at a right angle (90 degrees). They have slopes that are negative reciprocals of each other.
Q: How do I determine if two lines are parallel or perpendicular?
A: To determine if two lines are parallel or perpendicular, you need to compare their slopes. If the slopes are the same, the lines are parallel. If the slopes are negative reciprocals of each other, the lines are perpendicular.
Q: What is the slope of a line?
A: 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). It can be represented as a fraction, with the rise as the numerator and the run as the denominator.
Q: How do I find the point of intersection between two lines?
A: To find the point of intersection between two lines, you need to set the two equations equal to each other and solve for x. Once you have the value of x, you can substitute it into one of the original equations to find the value of y.
Q: What is the relationship between the slopes of parallel lines?
A: The slopes of parallel lines are the same. They have the same ratio of rise to run, but different y-intercepts.
Q: What is the relationship between the slopes of perpendicular lines?
A: The slopes of perpendicular lines are negative reciprocals of each other. They have slopes that are the negative of each other, multiplied by the reciprocal of each other.
Q: Can two lines be neither parallel nor perpendicular?
A: Yes, two lines can be neither parallel nor perpendicular. They can intersect at a point, but not at a right angle.
Q: How do I graph two lines on a coordinate plane?
A: To graph two lines on a coordinate plane, you need to plot the points of intersection and draw a line through them. You can use a ruler or a graphing tool to help you draw the lines.
Q: What is the significance of the point of intersection between two lines?
A: The point of intersection between two lines is the point where the two lines meet. It is an important concept in mathematics and is used in a variety of applications, including physics, engineering, and computer science.
Conclusion
In this article, we answered some frequently asked questions related to the topic of determining the relationship between two lines. We hope that this article has provided you with a better understanding of the concepts and has helped you to improve your skills in mathematics.
Additional Resources
For more information on determining the relationship between two lines, we recommend the following resources:
- Khan Academy: Linear Equations and Graphs
- Mathway: Linear Equations and Graphs
- Wolfram Alpha: Linear Equations and Graphs
Summary
In this article, we answered some frequently asked questions related to the topic of determining the relationship between two lines. We hope that this article has provided you with a better understanding of the concepts and has helped you to improve your skills in mathematics.