What Is The Equation Of A Line That Passes Through The Points \[$(3, 6)\$\] And \[$(8, 4)\$\]?A. \[$ Y = \frac{5}{2}x - \frac{3}{2} \$\]B. \[$ Y = \frac{5}{2}x - \frac{27}{2} \$\]C. \[$ Y = -\frac{2}{5}x -

by ADMIN 206 views

What is the Equation of a Line that Passes Through Two Given Points?

In mathematics, finding the equation of a line that passes through two given points is a fundamental concept in algebra and geometry. This concept is crucial in various fields, including physics, engineering, and computer science. In this article, we will explore the process of finding the equation of a line that passes through two given points, using the points {(3, 6)$}$ and {(8, 4)$}$ as examples.

Understanding the Problem

To find the equation of a line that passes through two given points, we need to use the concept of slope and the point-slope form of a linear equation. The slope of a line is a measure of how steep it is, and it can be calculated using the formula:

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

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

Calculating the Slope

Using the given points {(3, 6)$}$ and {(8, 4)$}$, we can calculate the slope of the line as follows:

m = (4 - 6) / (8 - 3) m = -2 / 5 m = -0.4

Finding the Equation of the Line

Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by:

y - y1 = m(x - x1)

where (x1, y1) is one of the given points, and m is the slope.

Using the point (3, 6) and the slope m = -0.4, we can write the equation of the line as:

y - 6 = -0.4(x - 3)

Simplifying the Equation

To simplify the equation, we can expand and combine like terms:

y - 6 = -0.4x + 1.2 y = -0.4x + 7.2

Comparing with the Options

Now that we have the equation of the line, we can compare it with the given options:

A. { y = \frac{5}{2}x - \frac{3}{2} $}$ B. { y = \frac{5}{2}x - \frac{27}{2} $}$ C. { y = -\frac{2}{5}x - \frac{18}{5} $}$

Comparing the equation we derived with the options, we can see that option C is the correct answer.

Conclusion

In conclusion, finding the equation of a line that passes through two given points involves calculating the slope and using the point-slope form of a linear equation. By following the steps outlined in this article, we can find the equation of a line that passes through any two given points. The correct answer is option C, { y = -\frac{2}{5}x - \frac{18}{5} $}$.

Additional Examples

Here are a few more examples of finding the equation of a line that passes through two given points:

  • Find the equation of the line that passes through the points (2, 3) and (4, 5).
  • Find the equation of the line that passes through the points (1, 2) and (3, 4).
  • Find the equation of the line that passes through the points (0, 0) and (2, 3).

Tips and Tricks

Here are a few tips and tricks to help you find the equation of a line that passes through two given points:

  • Make sure to calculate the slope correctly using the formula m = (y2 - y1) / (x2 - x1).
  • Use the point-slope form of a linear equation to find the equation of the line.
  • Simplify the equation by expanding and combining like terms.
  • Compare the equation you derived with the given options to find the correct answer.

Common Mistakes

Here are a few common mistakes to avoid when finding the equation of a line that passes through two given points:

  • Calculating the slope incorrectly using the formula m = (y2 - y1) / (x2 - x1).
  • Using the wrong point-slope form of a linear equation.
  • Not simplifying the equation by expanding and combining like terms.
  • Not comparing the equation you derived with the given options to find the correct answer.

Real-World Applications

Finding the equation of a line that passes through two given points has many real-world applications, including:

  • Physics: Finding the equation of a line that represents the motion of an object.
  • Engineering: Finding the equation of a line that represents the trajectory of a projectile.
  • Computer Science: Finding the equation of a line that represents the boundary of a shape.

Conclusion

In conclusion, finding the equation of a line that passes through two given points is a fundamental concept in algebra and geometry. By following the steps outlined in this article, we can find the equation of a line that passes through any two given points. The correct answer is option C, { y = -\frac{2}{5}x - \frac{18}{5} $}$.
Q&A: Finding the Equation of a Line that Passes Through Two Given Points

In the previous article, we explored the process of finding the equation of a line that passes through two given points. In this article, we will answer some frequently asked questions about finding the equation of a line that passes through two given points.

Q: What is the point-slope form of a linear equation?

A: The point-slope form of a linear equation is given by:

y - y1 = m(x - x1)

where (x1, y1) is one of the given points, and m is 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 = (y2 - y1) / (x2 - x1)

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

Q: What is the difference between the slope-intercept form and the point-slope form of a linear equation?

A: The slope-intercept form of a linear equation is given by:

y = mx + b

where m is the slope, and b is the y-intercept.

The point-slope form of a linear equation is given by:

y - y1 = m(x - x1)

where (x1, y1) is one of the given points, and m is the slope.

The main difference between the two forms is that the slope-intercept form requires you to know the y-intercept, while the point-slope form requires you to know one of the given points.

Q: Can I use the point-slope form to find the equation of a line that passes through three or more points?

A: Yes, you can use the point-slope form to find the equation of a line that passes through three or more points. However, you will need to find the slope of the line using the formula:

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

and then use the point-slope form to find the equation of the line.

Q: What if I have two points that are the same? Can I still find the equation of the line?

A: If you have two points that are the same, then the line is a vertical line, and you can find the equation of the line by using the x-coordinate of the point as the x-intercept.

Q: Can I use the point-slope form to find the equation of a line that passes through a point and a line?

A: Yes, you can use the point-slope form to find the equation of a line that passes through a point and a line. However, you will need to find the slope of the line using the formula:

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

and then use the point-slope form to find the equation of the line.

Q: What if I have a point and a line, but the point is not on the line? Can I still find the equation of the line?

A: If you have a point and a line, but the point is not on the line, then you cannot find the equation of the line using the point-slope form. However, you can find the equation of the line using other methods, such as finding the distance from the point to the line.

Q: Can I use the point-slope form to find the equation of a line that passes through a point and a circle?

A: Yes, you can use the point-slope form to find the equation of a line that passes through a point and a circle. However, you will need to find the slope of the line using the formula:

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

and then use the point-slope form to find the equation of the line.

Q: What if I have a point and a circle, but the point is not on the circle? Can I still find the equation of the line?

A: If you have a point and a circle, but the point is not on the circle, then you cannot find the equation of the line using the point-slope form. However, you can find the equation of the line using other methods, such as finding the distance from the point to the circle.

Conclusion

In conclusion, finding the equation of a line that passes through two given points is a fundamental concept in algebra and geometry. By following the steps outlined in this article, we can find the equation of a line that passes through any two given points. We also answered some frequently asked questions about finding the equation of a line that passes through two given points.

Additional Resources

Here are some additional resources that you may find helpful:

  • Khan Academy: Finding the Equation of a Line
  • Mathway: Finding the Equation of a Line
  • Wolfram Alpha: Finding the Equation of a Line

Practice Problems

Here are some practice problems that you can use to practice finding the equation of a line that passes through two given points:

  • Find the equation of the line that passes through the points (2, 3) and (4, 5).
  • Find the equation of the line that passes through the points (1, 2) and (3, 4).
  • Find the equation of the line that passes through the points (0, 0) and (2, 3).

Conclusion

In conclusion, finding the equation of a line that passes through two given points is a fundamental concept in algebra and geometry. By following the steps outlined in this article, we can find the equation of a line that passes through any two given points. We also answered some frequently asked questions about finding the equation of a line that passes through two given points.