Select The Correct Answer.What Is The Slope Of The Line That Goes Through The Points { (1, -5)$}$ And { (4, 1)$}$?A. { -\frac{4}{3}$}$B. { -\frac{3}{4}$}$C. { \frac{1}{2}$}$D. 2

by ADMIN 178 views

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 crucial concept in geometry, algebra, and calculus. In this article, we will explore how to calculate the slope of a line that passes through two given points.

What is Slope?

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 can be positive, negative, or zero, depending on the orientation of the line.

Calculating the Slope

To calculate the slope of a line that passes through two points, we can use the following formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Example: Calculating the Slope of a Line

Let's consider the two points (1, -5) and (4, 1). We want to find the slope of the line that passes through these two points.

First, we identify the coordinates of the two points:

(x1, y1) = (1, -5) (x2, y2) = (4, 1)

Next, we plug these values into the formula:

m = (y2 - y1) / (x2 - x1) = (1 - (-5)) / (4 - 1) = (1 + 5) / 3 = 6 / 3 = 2

Therefore, the slope of the line that passes through the points (1, -5) and (4, 1) is 2.

Choosing the Correct Answer

Now that we have calculated the slope of the line, we can compare it to the answer choices:

A. -4/3 B. -3/4 C. 1/2 D. 2

Based on our calculation, the correct answer is:

D. 2

Conclusion

In this article, we have learned how to calculate the slope of a line that passes through two given points. We have used the formula m = (y2 - y1) / (x2 - x1) and applied it to an example problem. We have also compared our answer to the answer choices and selected the correct one. With this knowledge, you can now calculate the slope of a line with ease.

Additional Tips and Tricks

  • Make sure to label the coordinates of the two points correctly.
  • Use the correct formula to calculate the slope.
  • Check your answer by plugging it back into the formula.
  • Practice, practice, practice! Calculating the slope of a line is a skill that requires practice to develop.

Common Mistakes to Avoid

  • Don't forget to label the coordinates of the two points correctly.
  • Don't use the wrong formula to calculate the slope.
  • Don't round your answer incorrectly.
  • Don't forget to check your answer by plugging it back into the formula.

Real-World Applications

Calculating the slope of a line has many real-world applications, including:

  • Physics: Calculating the slope of a line can help us understand the motion of objects.
  • Engineering: Calculating the slope of a line can help us design and build structures such as bridges and buildings.
  • Economics: Calculating the slope of a line can help us understand the relationship between variables such as supply and demand.

Conclusion

Q: What is the slope of a line?

A: 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).

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

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

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Q: What if the line is vertical?

A: If the line is vertical, the slope is undefined. This is because the line does not rise or fall over a given horizontal distance.

Q: What if the line is horizontal?

A: If the line is horizontal, the slope is 0. This is because the line does not rise or fall over a given horizontal distance.

Q: Can I use the slope formula to find the equation of a line?

A: Yes, you can use the slope formula to find the equation of a line. Once you have the slope, you can use the point-slope form of a line to find the equation.

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

A: To find the equation of a line using the slope formula, you can use the point-slope form of a line:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope.

Q: What if I have two points on a line and I want to find the equation of the line?

A: If you have two points on a line and you want to find the equation of the line, you can use the slope formula to find the slope and then use the point-slope form of a line to find the equation.

Q: Can I use the slope formula to find the distance between two points?

A: No, the slope formula is used to find the slope of a line, not the distance between two points. To find the distance between two points, you can use the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Q: What if I have a line with a negative slope?

A: If you have a line with a negative slope, it means that the line falls as you move to the right. This is because the slope is the ratio of the vertical change to the horizontal change, and a negative slope indicates that the vertical change is negative.

Q: Can I use the slope formula to find the midpoint of a line segment?

A: No, the slope formula is used to find the slope of a line, not the midpoint of a line segment. To find the midpoint of a line segment, you can use the midpoint formula:

(x1 + x2)/2, (y1 + y2)/2

Q: What if I have a line with a slope of 1?

A: If you have a line with a slope of 1, it means that the line rises 1 unit for every 1 unit of horizontal change. This is a special case, and the line is said to have a slope of 1.

Q: Can I use the slope formula to find the equation of a circle?

A: No, the slope formula is used to find the slope of a line, not the equation of a circle. To find the equation of a circle, you can use the general form of a circle:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) is the center of the circle and r is the radius.

Conclusion

In conclusion, the slope formula is a powerful tool for finding the slope of a line. By understanding how to use the slope formula, you can solve a wide range of problems involving lines and slopes. Remember to label the coordinates of the two points correctly, use the correct formula, and check your answer by plugging it back into the formula. With practice, you can become proficient in using the slope formula to solve problems involving lines and slopes.