Which Of The Following Slopes Of A Line Pass Through The Points ( 1 , − 3 (1, -3 ( 1 , − 3 ] And ( 0 , 2 (0, 2 ( 0 , 2 ]?A. M = 5 M = 5 M = 5 B. M = − 5 M = -5 M = − 5 C. M M M Is Undefined D. None Of These Choices Are Correct.

Mar 11, 2025 by ADMIN 229 views

**Slope of a Line: Understanding the Concept**

Introduction

In mathematics, the slope of a line is a fundamental concept that helps us understand the steepness or incline of a line. It is a measure of how much the line rises (or falls) vertically for every unit of horizontal distance traveled. In this article, we will explore the concept of slope and determine which of the given slopes pass through the points $(1, -3)$ and $(0, 2)$ .

What is Slope?

The slope of a line is denoted by the letter $m$ and is calculated using 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.

Calculating Slope

To calculate the slope of a line passing through two points, we can use the formula above. Let's calculate the slope of the line passing through the points $(1, -3)$ and $(0, 2)$ .

m = \frac{2 - (-3)}{0 - 1} = \frac{5}{-1} = -5

Understanding the Options

Now that we have calculated the slope of the line passing through the points $(1, -3)$ and $(0, 2)$ , let's examine the options given:

A. $m = 5$ B. $m = -5$ C. $m$ is undefined D. None of these choices are correct

Analyzing Option A

Option A states that the slope of the line is $m = 5$ . However, we have calculated the slope to be $m = -5$ , not $m = 5$ . Therefore, option A is incorrect.

Analyzing Option B

Option B states that the slope of the line is $m = -5$ . We have calculated the slope to be $m = -5$ , which matches option B. Therefore, option B is correct.

Analyzing Option C

Option C states that the slope of the line is undefined. However, we have calculated the slope to be $m = -5$ , which is a finite value. Therefore, option C is incorrect.

Analyzing Option D

Option D states that none of the choices are correct. However, we have determined that option B is correct, which means that option D is incorrect.

Conclusion

In conclusion, the slope of the line passing through the points $(1, -3)$ and $(0, 2)$ is $m = -5$ . Therefore, the correct answer is option B.

Additional Examples

To further illustrate the concept of slope, let's consider a few more examples.

Example 1

Find the slope of the line passing through the points $(2, 4)$ and $(3, 6)$ .

m = \frac{6 - 4}{3 - 2} = \frac{2}{1} = 2

Example 2

Find the slope of the line passing through the points $(0, 0)$ and $(1, 1)$ .

m = \frac{1 - 0}{1 - 0} = \frac{1}{1} = 1

Example 3

Find the slope of the line passing through the points $(1, 2)$ and $(2, 3)$ .

m = \frac{3 - 2}{2 - 1} = \frac{1}{1} = 1

Final Thoughts

In conclusion, the concept of slope is a fundamental aspect of mathematics that helps us understand the steepness or incline of a line. By calculating the slope of a line passing through two points, we can determine the steepness of the line. In this article, we have explored the concept of slope and determined which of the given slopes pass through the points $(1, -3)$ and $(0, 2)$ . We have also provided additional examples to further illustrate the concept of slope.

References

[1] Khan Academy. (n.d.). Slope of a line. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f7f/slope-of-a-line/v/slope-of-a-line
[2] Math Open Reference. (n.d.). Slope of a line. Retrieved from https://www.mathopenref.com/slope.html
[3] Purplemath. (n.d.). Slope of a line. Retrieved from https://www.purplemath.com/modules/slope.htm
Slope of a Line: Q&A =========================

Introduction

In our previous article, we explored the concept of slope and determined which of the given slopes pass through the points $(1, -3)$ and $(0, 2)$ . In this article, we will answer some frequently asked questions about the slope of a line.

Q&A

Q: What is the slope of a horizontal line?

A: The slope of a horizontal line is 0. This is because the line does not rise or fall vertically, and the change in y-coordinates is 0.

Q: What is the slope of a vertical line?

A: The slope of a vertical line is undefined. This is because the line does not change horizontally, and the change in x-coordinates is 0.

Q: How do I calculate the slope of a line passing through two points?

A: To calculate the slope of a line passing through two points, 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 the two points on the line.

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

A: A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right.

Q: Can a line have a slope of 0?

A: Yes, a line can have a slope of 0. This is the case for a horizontal line.

Q: Can a line have an undefined slope?

A: Yes, a line can have an undefined slope. This is the case for a vertical line.

Q: How do I determine the slope of a line from its graph?

A: To determine the slope of a line from its graph, you can use the following steps:

Choose two points on the line.
Calculate the change in y-coordinates (rise) and the change in x-coordinates (run).
Divide the rise by the run to get the slope.

Q: What is the significance of the slope of a line?

A: The slope of a line is significant because it helps us understand the steepness or incline of the line. It is also used in various mathematical and real-world applications, such as calculating the rate of change of a function.