Write The Equation Of The Line Passing Through The Points (8, -9) And (5, 2) In Point-slope Form.

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Introduction

In mathematics, the equation of a line can be expressed in various forms, including slope-intercept form, standard form, and point-slope form. The point-slope form is a popular method for finding the equation of a line when two points on the line are given. In this article, we will explore how to write the equation of a line passing through the points (8, -9) and (5, 2) in point-slope form.

What is Point-Slope Form?

Point-slope form is a way of expressing the equation of a line using the coordinates of a point on the line and the slope of the line. The general form of a point-slope equation is:

y - y1 = m(x - x1)

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

Finding the Slope

To find the equation of a line in point-slope form, we need to find the slope of the line first. The slope of a line can be calculated using the formula:

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

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

Calculating the Slope

Using the given points (8, -9) and (5, 2), we can calculate the slope of the line as follows:

m = (2 - (-9)) / (5 - 8) m = (2 + 9) / (-3) m = 11 / (-3) m = -11/3

Writing the Equation in Point-Slope Form

Now that we have the slope, we can write the equation of the line in point-slope form using the coordinates of one of the points. Let's use the point (8, -9).

y - (-9) = (-11/3)(x - 8)

Simplifying the Equation

To simplify the equation, we can multiply both sides by 3 to eliminate the fraction:

3(y + 9) = -11(x - 8)

Expanding the equation, we get:

3y + 27 = -11x + 88

Rearranging the Equation

To put the equation in a more familiar form, we can rearrange it to isolate y:

3y = -11x + 88 - 27 3y = -11x + 61

Dividing both sides by 3, we get:

y = (-11/3)x + 61/3

Conclusion

In this article, we have learned how to write the equation of a line passing through the points (8, -9) and (5, 2) in point-slope form. We first calculated the slope of the line using the formula m = (y2 - y1) / (x2 - x1). Then, we used the coordinates of one of the points and the slope to write the equation in point-slope form. Finally, we simplified and rearranged the equation to put it in a more familiar form.

Key Takeaways

  • The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
  • The slope of a line can be calculated using the formula m = (y2 - y1) / (x2 - x1).
  • To write the equation of a line in point-slope form, use the coordinates of one of the points and the slope.
  • Simplify and rearrange the equation to put it in a more familiar form.

Practice Problems

  1. Find the equation of the line passing through the points (2, 3) and (4, 5) in point-slope form.
  2. Write the equation of the line passing through the points (1, -2) and (3, 4) in point-slope form.
  3. Find the equation of the line passing through the points (0, 2) and (2, 4) in point-slope form.

Answer Key

  1. y - 3 = (1/2)(x - 2)
  2. y - (-2) = (1/2)(x - 3)
  3. y - 2 = (1)(x - 0)

References

Q: What is point-slope form?

A: Point-slope form is a way of expressing the equation of a line using the coordinates of a point on the line and the slope of the line. The general form of a point-slope equation is:

y - y1 = m(x - x1)

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

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 two points on the line.

Q: What if I have a point and the slope, but not the other point?

A: If you have a point and the slope, but not the other point, you can use the point-slope form to write the equation of the line. For example, if you have the point (2, 3) and the slope m = 1/2, you can write the equation as:

y - 3 = (1/2)(x - 2)

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

A: Yes, you can use point-slope form to find the equation of a horizontal line. A horizontal line has a slope of 0, so the equation will be in the form:

y - y1 = 0(x - x1)

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

A: Yes, you can use point-slope form to find the equation of a vertical line. A vertical line has an undefined slope, so the equation will be in the form:

x - x1 = 0

Q: How do I simplify a point-slope equation?

A: To simplify a point-slope equation, you can multiply both sides by the denominator to eliminate the fraction. For example, if you have the equation:

y - 3 = (1/2)(x - 2)

you can multiply both sides by 2 to get:

2(y - 3) = (x - 2)

Expanding the equation, you get:

2y - 6 = x - 2

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

A: Yes, you can use point-slope form to find the equation of a line that passes through the origin. If the line passes through the origin, the point (0, 0) is on the line. You can use this point and the slope to write the equation of the line.

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

A: Yes, you can use point-slope form to find the equation of a line that is parallel to another line. If two lines are parallel, they have the same slope. You can use the slope of the first line and a point on the second line to write the equation of the second line.

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

A: Yes, you can use point-slope form to find the equation of a line that is perpendicular to another line. If two lines are perpendicular, their slopes are negative reciprocals of each other. You can use the slope of the first line and a point on the second line to write the equation of the second line.

Conclusion

In this article, we have answered some frequently asked questions about point-slope form. We have covered topics such as finding the slope of a line, using point-slope form to find the equation of a line, and simplifying point-slope equations. We hope that this article has been helpful in understanding point-slope form and how to use it to find the equation of a line.