Determining the Slope of a Parallel Line

Oct 13, 2025 by magicalceylontours.com 52 views

Introduction

In the standard ${$ (x, y)$}$ coordinate plane, the slope of a line is a fundamental concept in mathematics that describes the steepness or incline of the line. It is a crucial concept in geometry, algebra, and calculus, and is used to determine the relationship between two variables. In this article, we will explore the concept of the slope of a line and determine the slope of a line that is parallel to the equation ${$ x + 5y = 9$}$.

What is the Slope of a Line?

The slope of a line is a measure of how much the line rises (or falls) vertically over a given horizontal distance. It is denoted by the letter ${$ m$}$ and is calculated as the ratio of the vertical change (rise) to the horizontal change (run). The slope of a line can be calculated using the following formula:

${$ m = \frac{y_2 - y_1}{x_2 - x_1}$}$

where ${$ y_2 - y_1$}$ is the vertical change and ${$ x_2 - x_1$}$ is the horizontal change.

The Equation of a Line

The equation of a line in the standard ${$ (x, y)$}$ coordinate plane can be written in the form:

${$ Ax + By = C$}$

where ${$ A$}$, ${$ B$}$, and ${$ C$}$ are constants. The equation of a line can also be written in the slope-intercept form:

${$ y = mx + b$}$

where ${$ m$}$ is the slope of the line and ${$ b$}$ is the y-intercept.

Parallel Lines

Two lines are said to be parallel if they have the same slope. In other words, if two lines have the same slope, they will never intersect, and will always remain a fixed distance apart.

Determining the Slope of a Parallel Line

To determine the slope of a line that is parallel to the equation ${$ x + 5y = 9$}$, we need to first rewrite the equation in the slope-intercept form. We can do this by isolating the variable ${$ y$}$ on one side of the equation.

${$ x + 5y = 9$}$

${ $5y = -x + 9\$}$

${$ y = -\frac{1}{5}x + \frac{9}{5}$}$

Now that we have the equation of the line in the slope-intercept form, we can see that the slope of the line is ${$ -\frac{1}{5}$}$.

Conclusion

In conclusion, the slope of a line that is parallel to the equation ${$ x + 5y = 9$}$ is ${$ -\frac{1}{5}$}$. This is because the slope of a line is a measure of how much the line rises (or falls) vertically over a given horizontal distance, and two lines are said to be parallel if they have the same slope.

Answer

The correct answer is:

${$ B$}$

${$ -\frac{1}{5}$}$

Additional Resources

For more information on the slope of a line, please refer to the following resources:

Discussion

Frequently Asked Questions

Q: What is the slope of a line that is parallel to the equation ${$ x + 5y = 9$}$? A: The slope of a line that is parallel to the equation ${$ x + 5y = 9$}$ is ${$ -\frac{1}{5}$}$.

Q: How do I determine the slope of a line that is parallel to a given equation? A: To determine the slope of a line that is parallel to a given equation, you need to first rewrite the equation in the slope-intercept form. Then, you can see the slope of the line.

Q: What is the difference between the slope of a line and the slope of a parallel line? A: The slope of a line is a measure of how much the line rises (or falls) vertically over a given horizontal distance. The slope of a parallel line is the same as the slope of the original line.

Q: Can two lines have the same slope but not be parallel? A: No, two lines cannot have the same slope but not be parallel. If two lines have the same slope, they will always be parallel.

**Q: How do I calculate the slope of a line using the formula [ $m = \frac{y_2 - y_1}{x_2 - x_1}\$]?** A: To calculate the slope of a line using the formula \[$ m = \frac{y_2 - y_1}{x_2 - x_1}$], you need to know the coordinates of two points on the line. Then, you can plug in the values into the formula and calculate the slope.

Q: What is the significance of the slope of a line in real-world applications? A: The slope of a line is used in many real-world applications, such as:

Calculating the rate of change of a quantity over time
Determining the steepness of a roof or a slope
Calculating the angle of a ramp or a staircase
Determining the rate of change of a population over time

Q: Can the slope of a line be negative? A: Yes, the slope of a line can be negative. A negative slope indicates that the line is falling or decreasing.

Q: Can the slope of a line be zero? A: Yes, the slope of a line can be zero. A zero slope indicates that the line is horizontal.

Q: Can the slope of a line be undefined? A: Yes, the slope of a line can be undefined. An undefined slope indicates that the line is vertical.

Conclusion

In conclusion, the slope of a line is a fundamental concept in mathematics that describes the steepness or incline of the line. It is used in many real-world applications and is a crucial concept in geometry, algebra, and calculus. We hope that this Q&A article has helped to clarify any questions or doubts you may have had about the slope of a line.

Additional Resources

For more information on the slope of a line, please refer to the following resources:

Discussion

Do you have any questions or comments about the slope of a line? Please feel free to share your thoughts and opinions in the comments section below.