Determine The Equation Of The Line That Passes Through The Given Points: (2, 6) And (4, 16).A. { Y = -4x + 5 $}$ B. { Y = 5x - 16 $}$ C. { Y = 5x - 4 $}$ D. { Y = -5x + 4 $}$Please Select The Best Answer

by ADMIN 207 views

Introduction

In mathematics, the equation of a line can be determined using various methods, including the slope-intercept form and the point-slope form. The slope-intercept form is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. The point-slope form is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. In this article, we will determine the equation of a line that passes through two given points.

The Problem

We are given two points: (2, 6) and (4, 16). We need to determine the equation of the line that passes through these two points.

Step 1: Find the Slope

To find the slope of the line, we can use the formula:

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

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

Plugging in the values, we get:

m = (16 - 6) / (4 - 2) m = 10 / 2 m = 5

Step 2: Find the y-Intercept

Now that we have the slope, we can use either of the two points to find the y-intercept. Let's use the point (2, 6).

We can plug in the values into the equation y = mx + b:

6 = 5(2) + b 6 = 10 + b b = -4

Step 3: Write the Equation of the Line

Now that we have the slope and the y-intercept, we can write the equation of the line in slope-intercept form:

y = mx + b y = 5x - 4

Conclusion

Therefore, the equation of the line that passes through the points (2, 6) and (4, 16) is y = 5x - 4.

Answer

The correct answer is:

C. { y = 5x - 4 $}$

Discussion

This problem is a classic example of how to determine the equation of a line passing through two points. The slope-intercept form is a useful tool for finding the equation of a line, and it is essential to understand how to use it to solve problems like this one.

Tips and Tricks

  • Make sure to use the correct formula for finding the slope.
  • Use either of the two points to find the y-intercept.
  • Write the equation of the line in slope-intercept form.

Related Topics

  • Slope-intercept form
  • Point-slope form
  • Equation of a line
  • Slope
  • Y-intercept

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Additional Resources

  • Khan Academy: Equation of a line
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram Alpha: Equation of a line
    Determining the Equation of a Line Passing Through Two Points: Q&A ====================================================================

Introduction

In our previous article, we discussed how to determine the equation of a line passing through two points using the slope-intercept form. In this article, we will answer some common questions related to this topic.

Q: What is the slope-intercept form of a line?

A: The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.

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

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

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

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

Q: How do I find the y-intercept of a line?

A: To find the y-intercept of a line, you can use either of the two points and plug in the values into the equation y = mx + b.

Q: What if the two points are not on the same line?

A: If the two points are not on the same line, then there is no equation of the line that passes through both points.

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

A: Yes, you can use the point-slope form to find the equation of a line. The point-slope form is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Q: How do I convert the point-slope form to the slope-intercept form?

A: To convert the point-slope form to the slope-intercept form, you can simplify the equation by distributing the slope and combining like terms.

Q: What are some common mistakes to avoid when finding the equation of a line?

A: Some common mistakes to avoid when finding the equation of a line include:

  • Not using the correct formula for finding the slope
  • Not using either of the two points to find the y-intercept
  • Not simplifying the equation when converting from point-slope form to slope-intercept form

Q: What are some real-world applications of finding the equation of a line?

A: Some real-world applications of finding the equation of a line include:

  • Modeling the growth of a population
  • Describing the motion of an object
  • Finding the equation of a circle or an ellipse

Conclusion

In this article, we answered some common questions related to determining the equation of a line passing through two points. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of this topic.

Additional Resources

  • Khan Academy: Equation of a line
  • MIT OpenCourseWare: Linear Algebra
  • Wolfram Alpha: Equation of a line

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Linear Algebra" by Jim Hefferon

Discussion

This article is a continuation of our previous article on determining the equation of a line passing through two points. We hope that this article has been helpful in providing a better understanding of this topic and answering some common questions. If you have any further questions or need additional clarification, please don't hesitate to ask.