Finding the Negative Reciprocal

Oct 13, 2025 by miodeya.com 3 views

Introduction

In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. 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. In this article, we will discuss the equation of line p and the slope of line q, which is perpendicular to line p.

The Equation of Line p

The equation of line p is given as y = (8/7)x + (1/6). This equation represents a linear relationship between the variables x and y. The slope of line p is 8/7, which means that for every unit increase in x, the value of y increases by 8/7 units.

The Slope of Line q

Line q is perpendicular to line p, which means that the slope of line q is the negative reciprocal of the slope of line p. To find the slope of line q, we need to find the negative reciprocal of 8/7.

Finding the Negative Reciprocal

The negative reciprocal of a number is obtained by multiplying the number by -1 and then taking the reciprocal. In this case, we need to multiply 8/7 by -1 and then take the reciprocal.

-1 × (8/7) = -8/7

To find the reciprocal of -8/7, we need to invert the fraction and change the sign.

1/(-8/7) = -7/8

Therefore, the slope of line q is -7/8.

Conclusion

In conclusion, the equation of line p is y = (8/7)x + (1/6), and the slope of line q is -7/8. The slope of line q is the negative reciprocal of the slope of line p, which is a fundamental property of perpendicular lines.

The Importance of Slope in Mathematics

The slope of a line is an important concept in mathematics that is used to describe the relationship between two variables. The slope of a line can be used to determine the direction and steepness of the line, and it is used in a variety of mathematical applications, including algebra, geometry, and calculus.

Real-World Applications of Slope

The concept of slope is used in a variety of real-world applications, including:

Physics: The slope of a line is used to describe the motion of objects, including the acceleration and velocity of objects.
Engineering: The slope of a line is used to design and build structures, including bridges and buildings.
Economics: The slope of a line is used to describe the relationship between two variables, including the demand and supply of goods and services.

Solving Problems Involving Slope

To solve problems involving slope, we need to use the concept of slope and the equation of a line. Here are some steps to follow:

Read the problem: Read the problem carefully and understand what is being asked.
Identify the variables: Identify the variables involved in the problem and determine the relationship between them.
Write the equation: Write the equation of the line using the variables and the slope.
Solve the equation: Solve the equation to find the value of the variables.
Check the solution: Check the solution to make sure it is correct.

Example Problems

Here are some example problems involving slope:

Problem 1: Find the slope of the line y = (3/4)x + (1/2).
Problem 2: Find the slope of the line y = (-2/3)x + (1/4).
Problem 3: Find the slope of the line y = (5/6)x + (1/3).

Solution to Problem 1

To find the slope of the line y = (3/4)x + (1/2), we need to identify the slope of the line, which is 3/4.

Solution to Problem 2

To find the slope of the line y = (-2/3)x + (1/4), we need to identify the slope of the line, which is -2/3.

Solution to Problem 3

To find the slope of the line y = (5/6)x + (1/3), we need to identify the slope of the line, which is 5/6.

Conclusion

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

Q: How do I find the slope of a line?

A: To find the slope of a line, you can use the formula: slope = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: What is the negative reciprocal of a slope?

A: The negative reciprocal of a slope is the slope of a line that is perpendicular to the original line. To find the negative reciprocal of a slope, you can multiply the slope by -1 and then take the reciprocal.

Q: What is the difference between a positive and negative slope?

A: A positive slope indicates that the line is rising from left to right, while a negative slope indicates that the line is falling from left to right.

Q: Can a line have a slope of zero?

A: Yes, a line can have a slope of zero. This occurs when the line is horizontal, meaning that it does not rise or fall at all.

Q: Can a line have an undefined slope?

A: Yes, a line can have an undefined slope. This occurs when the line is vertical, meaning that it does not have a horizontal change (run).

Q: How do I use the slope-intercept form of a line?

A: The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. To use this form, you can plug in the values of m and b to find the equation of the line.

Q: What is the y-intercept of a line?

A: The y-intercept of a line is the point where the line intersects the y-axis. It is the value of y when x is equal to zero.

Q: Can a line have multiple y-intercepts?

A: No, a line can only have one y-intercept.

Q: How do I graph a line using its slope and y-intercept?

A: To graph a line using its slope and y-intercept, you can use the slope-intercept form of the line (y = mx + b) and plot the y-intercept on the y-axis. Then, you can use the slope to determine the direction and steepness of the line.

Q: What is the relationship between the slope and the y-intercept of a line?

A: The slope and the y-intercept of a line are related in that the slope determines the direction and steepness of the line, while the y-intercept determines the point where the line intersects the y-axis.

Q: Can a line have a slope of 1 or -1?

A: Yes, a line can have a slope of 1 or -1. This occurs when the line is a 45-degree angle, meaning that it rises or falls at a rate of 1 unit for every 1 unit of horizontal change.

Q: What is the significance of the slope of a line in real-world applications?

A: The slope of a line is significant in real-world applications because it can be used to model and analyze the behavior of physical systems, such as the motion of objects, the flow of fluids, and the growth of populations.

Q: Can a line have a slope that is a fraction or a decimal?

A: Yes, a line can have a slope that is a fraction or a decimal. This occurs when the line is not a straight line, but rather a curved line that has a non-integer slope.

Q: How do I convert a fraction or decimal slope to a percentage?

A: To convert a fraction or decimal slope to a percentage, you can multiply the slope by 100 and then add a percentage sign (%). For example, a slope of 0.5 can be converted to a percentage as 50%.