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Oct 13, 2025 by windailys.com 30 views

What is the Slope of a Line Parallel to $y=\frac{3}{4} x+2$ ?

Understanding Slope and Parallel Lines

In mathematics, the slope of a line is a fundamental concept used to describe the steepness or incline of the line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope of a line is denoted by the letter 'm' and is calculated using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Parallel lines are lines that lie in the same plane and never intersect, no matter how far they are extended. One of the key properties of parallel lines is that they have the same slope. This means that if two lines are parallel, their slopes are equal.

The Slope of the Given Line

The given line is represented by the equation y = (3/4)x + 2. To find the slope of this line, we can rewrite the equation in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope of the given line is 3/4.

Finding the Slope of a Parallel Line

Since parallel lines have the same slope, we can conclude that the slope of a line parallel to the given line is also 3/4. However, the question asks for the slope of a line that is parallel to the given line, but with a negative slope. This means that the slope of the parallel line is the negative reciprocal of the slope of the given line.

Calculating the Negative Reciprocal

The negative reciprocal of a number is obtained by taking the reciprocal of the number and changing its sign. In this case, the reciprocal of 3/4 is 4/3, and the negative reciprocal is -4/3.

Conclusion

Based on the properties of parallel lines and the concept of slope, we can conclude that the slope of a line that is parallel to the line y = (3/4)x + 2 is -4/3.

Answer

The correct answer is A. -4/3.

Additional Examples and Applications

Parallel lines have numerous applications in mathematics, physics, and engineering. Some examples include:

Geometry: Parallel lines are used to define the properties of angles and shapes.
Physics: Parallel lines are used to describe the motion of objects and the forces acting upon them.
Engineering: Parallel lines are used to design and build structures such as bridges, buildings, and roads.

Real-World Applications

Parallel lines have numerous real-world applications, including:

Architecture: Parallel lines are used to design and build buildings, bridges, and other structures.
Engineering: Parallel lines are used to design and build machines, mechanisms, and systems.
Physics: Parallel lines are used to describe the motion of objects and the forces acting upon them.

Conclusion

In conclusion, the slope of a line that is parallel to the line y = (3/4)x + 2 is -4/3. This is based on the properties of parallel lines and the concept of slope. Parallel lines have numerous applications in mathematics, physics, and engineering, and are used to describe the properties of angles and shapes, the motion of objects, and the forces acting upon them.
Q&A: Understanding the Slope of Parallel Lines

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about the slope of parallel lines.

Q: What is the slope of a line that is parallel to the line y = (3/4)x + 2?

A: The slope of a line that is parallel to the line y = (3/4)x + 2 is -4/3.

Q: Why do parallel lines have the same slope?

A: Parallel lines have the same slope because they lie in the same plane and never intersect, no matter how far they are extended. This means that the slope of a line is a unique property that distinguishes it from other lines.

Q: How do I find the slope of a line that is parallel to a given line?

A: To find the slope of a line that is parallel to a given line, you can use the following steps:

Find the slope of the given line.
Take the negative reciprocal of the slope.
The resulting slope is the slope of a line that is parallel to the given line.

Q: What is the negative reciprocal of a number?

A: The negative reciprocal of a number is obtained by taking the reciprocal of the number and changing its sign. For example, the negative reciprocal of 3/4 is -4/3.

Q: Why is it important to understand the slope of parallel lines?

A: Understanding the slope of parallel lines is important because it helps you to:

Describe the properties of angles and shapes: Parallel lines are used to define the properties of angles and shapes.
Describe the motion of objects: Parallel lines are used to describe the motion of objects and the forces acting upon them.
Design and build structures: Parallel lines are used to design and build structures such as bridges, buildings, and roads.

Q: What are some real-world applications of parallel lines?

A: Some real-world applications of parallel lines include:

Architecture: Parallel lines are used to design and build buildings, bridges, and other structures.
Engineering: Parallel lines are used to design and build machines, mechanisms, and systems.
Physics: Parallel lines are used to describe the motion of objects and the forces acting upon them.

Q: How do I apply the concept of slope to real-world problems?

A: To apply the concept of slope to real-world problems, you can use the following steps:

Identify the problem and the relevant information.
Use the concept of slope to describe the properties of the problem.
Use the concept of slope to solve the problem.

Q: What are some common mistakes to avoid when working with parallel lines?

A: Some common mistakes to avoid when working with parallel lines include:

Confusing the slope with the y-intercept: The slope and the y-intercept are two different properties of a line.
Not considering the negative reciprocal: The negative reciprocal of a number is an important concept when working with parallel lines.
Not applying the concept of slope to real-world problems: The concept of slope has numerous applications in real-world problems.

Q: How do I practice and improve my understanding of the slope of parallel lines?

A: To practice and improve your understanding of the slope of parallel lines, you can use the following steps:

Practice solving problems that involve parallel lines.
Review the concept of slope and its applications.
Apply the concept of slope to real-world problems.

By following these steps, you can improve your understanding of the slope of parallel lines and apply it to real-world problems.