What Is The Slope Of A Line, In The Standard { (x, Y)$}$ Coordinate Plane, That Is Parallel To { X + 5y = 9$}$?A. { -5$}$ B. { -\frac{1}{5}$}$ C. { \frac{1}{5}$}$ D. { \frac{9}{5}$}$ E.

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Introduction

In the standard {(x, y)$}$ coordinate plane, the slope of a line is a fundamental concept in mathematics that helps us understand the steepness or incline of a line. The slope is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In this article, we will explore the concept of slope 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 steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope is denoted by the letter {m$}$ and can be calculated using the following formula:

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

where {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are two points on the line.

Parallel Lines

Parallel lines are lines that lie in the same plane and never intersect. In the coordinate plane, parallel lines have the same slope. If two lines are parallel, then their slopes are equal.

The Equation of a Line

The equation of a line in the coordinate plane can be written in the form {y = mx + b$}$, where {m$}$ is the slope of the line and {b$}$ is the y-intercept. The y-intercept is the point where the line intersects the y-axis.

Finding the Slope of a Line Parallel to {x + 5y = 9$}$

To find the slope of a line parallel to {x + 5y = 9$}$, we need to rewrite the equation in the form {y = mx + b$}$. We can do this by isolating {y$}$ on one side of the equation.

{x + 5y = 9$}$

Subtracting {x$}$ from both sides:

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

Dividing both sides by ${5\$}:

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

Now that we have the equation of the line in the form {y = mx + b$}$, 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 {x + 5y = 9$}$ is {-\frac{1}{5}$}$. This is because parallel lines have the same slope, and the equation of the line can be rewritten in the form {y = mx + b$}$ to find the slope.

Answer

The correct answer is:

  • B. {-\frac{1}{5}$}$

Discussion

What is the slope of a line that is parallel to {x + 5y = 9$}$? Do you have any questions about the concept of slope or parallel lines? Share your thoughts and questions in the comments below!

Related Topics

  • Slope of a Line: Learn more about the concept of slope and how to calculate it.
  • Parallel Lines: Explore the concept of parallel lines and how to identify them in the coordinate plane.
  • Equation of a Line: Learn more about the equation of a line and how to rewrite it in the form {y = mx + b$}$.

References

  • Math Open Reference: A comprehensive online reference for mathematics.
  • Khan Academy: A free online learning platform that offers courses and resources on mathematics and other subjects.
  • Wolfram Alpha: A powerful online calculator that can be used to solve mathematical problems and explore mathematical concepts.
    Frequently Asked Questions (FAQs) About the Slope of a Line ================================================================

Q: What is the slope of a line that is parallel to {x + 5y = 9$}$?

A: The slope of a line that is parallel to {x + 5y = 9$}$ is {-\frac{1}{5}$}$. This is because parallel lines have the same slope, and the equation of the line can be rewritten in the form {y = mx + b$}$ to find the slope.

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

A: To calculate the slope of a line, you can use the formula:

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

where {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are two points on the line.

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

A: The slope of a line is a measure of how steep it is, while the y-intercept is the point where the line intersects the y-axis. The slope is denoted by the letter {m$}$ and can be calculated using the formula above, while the y-intercept is denoted by the letter {b$}$ and can be found by substituting {x = 0$}$ into the equation of the line.

Q: Can two lines have the same slope but different y-intercepts?

A: Yes, two lines can have the same slope but different y-intercepts. For example, the lines {y = 2x + 3$}$ and {y = 2x + 5$}$ have the same slope (${2\$}) but different y-intercepts (${3\$} and ${5\$}, respectively).

Q: How do I determine if two lines are parallel?

A: To determine if two lines are parallel, you can compare their slopes. If the slopes are equal, then the lines are parallel.

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 as you move along it.

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 defined slope.

Q: How do I graph a line with a given slope and y-intercept?

A: To graph a line with a given slope and y-intercept, you can use the slope-intercept form of the equation of a line ({y = mx + b$}$) and plot the point where the line intersects the y-axis. Then, use the slope to determine the direction of the line and plot additional points as needed.

Q: Can I use the slope of a line to determine its equation?

A: Yes, you can use the slope of a line to determine its equation. If you know the slope and a point on the line, you can use the point-slope form of the equation of a line ({y - y_1 = m(x - x_1)$}$) to find the equation of the line.

Q: How do I use the slope of a line to solve a problem?

A: To use the slope of a line to solve a problem, you can apply the concept of slope to the problem at hand. For example, if you are given the slope of a line and a point on the line, you can use the point-slope form of the equation of a line to find the equation of the line and solve the problem.

Conclusion

In conclusion, the slope of a line is a fundamental concept in mathematics that helps us understand the steepness or incline of a line. By understanding the concept of slope and how to calculate it, you can apply it to a variety of problems and situations. Whether you are graphing a line, determining its equation, or solving a problem, the slope of a line is an essential tool to have in your mathematical toolkit.