Find The Equation Of The Line Using The Point-slope Formula. Write The Final Equation Using The Slope-intercept Form. Given Points: $(x_1, Y_1) = (-6, 10$\] And $(x_2, Y_2) = (7, -3$\].Recall That The Slope Of A Line Passing Through The

Feb 27, 2025 by ADMIN 237 views

Introduction

The point-slope formula is a fundamental concept in mathematics, particularly in algebra and geometry. It is used to find the equation of a line that passes through two given points. In this article, we will explore the point-slope formula, its application, and how to write the final equation in slope-intercept form.

The Point-Slope Formula

The point-slope formula is given by:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line, and m is the slope of the line. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

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

Given Points

We are given two points: (x1, y1) = (-6, 10) and (x2, y2) = (7, -3). We will use these points to find the slope of the line and then use the point-slope formula to find the equation of the line.

Finding the Slope

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

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

Substituting the given values, we get:

m = (-3 - 10) / (7 - (-6)) m = (-13) / (13) m = -1

Writing the Equation in Point-Slope Form

Now that we have the slope, we can use the point-slope formula to write the equation of the line. We will use the point (x1, y1) = (-6, 10) and the slope m = -1.

y - 10 = -1(x - (-6)) y - 10 = -1(x + 6) y - 10 = -x - 6 y = -x - 16

Writing the Equation in Slope-Intercept Form

The slope-intercept form of a line is given by:

y = mx + b

where m is the slope, and b is the y-intercept. We have already found the slope (m = -1) and the equation of the line in point-slope form (y = -x - 16). To write the equation in slope-intercept form, we need to isolate the y-variable.

y = -x - 16 y = -x + (-16) y = -x - 16

Conclusion

In this article, we have used the point-slope formula to find the equation of a line passing through two given points. We have also written the final equation in slope-intercept form. The point-slope formula is a powerful tool in mathematics, and its application is essential in various fields, including algebra, geometry, and calculus.

Key Takeaways

The point-slope formula is given by: y - y1 = m(x - x1)
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by: m = (y2 - y1) / (x2 - x1)
To write the equation of a line in slope-intercept form, we need to isolate the y-variable.

Frequently Asked Questions

What is the point-slope formula? The point-slope formula is a formula used to find the equation of a line that passes through two given points.
How do I find the slope of a line passing through two points? To find the slope of a line passing through two points, you can use the formula: m = (y2 - y1) / (x2 - x1)
How do I write the equation of a line in slope-intercept form? To write the equation of a line in slope-intercept form, you need to isolate the y-variable.

References

[1] "Point-Slope Formula." Math Open Reference, mathopenref.com/point-slope.html.
[2] "Slope-Intercept Form." Math Is Fun, mathisfun.com/algebra/slope-intercept-form.html.

Additional Resources

Khan Academy: Point-Slope Formula
Mathway: Point-Slope Formula
Wolfram Alpha: Point-Slope Formula

Introduction

The point-slope formula is a fundamental concept in mathematics, particularly in algebra and geometry. It is used to find the equation of a line that passes through two given points. In this article, we will answer some of the most frequently asked questions about the point-slope formula.

Q&A

Q1: What is the point-slope formula?

A1: The point-slope formula is a formula used to find the equation of a line that passes through two given points. It is given by: y - y1 = m(x - x1), where (x1, y1) is a point on the line, and m is the slope of the line.

Q2: How do I find the slope of a line passing through two points?

A2: To find the slope of a line passing through two points, you can use the formula: m = (y2 - y1) / (x2 - x1). This formula calculates the slope of the line by dividing the difference in y-coordinates by the difference in x-coordinates.

Q3: What is the slope-intercept form of a line?

A3: The slope-intercept form of a line is given by: y = mx + b, where m is the slope of the line, and b is the y-intercept. This form is used to write the equation of a line in a more simplified way.

Q4: How do I write the equation of a line in slope-intercept form?

A4: To write the equation of a line in slope-intercept form, you need to isolate the y-variable. This can be done by rearranging the equation in point-slope form to get: y = mx + b.

Q5: What is the y-intercept of a line?

A5: The y-intercept of a line is the point where the line intersects the y-axis. It is denoted by the letter b in the slope-intercept form of a line.

Q6: How do I find the y-intercept of a line?

A6: To find the y-intercept of a line, you can use the point-slope formula and set x = 0. This will give you the value of y, which is the y-intercept.

Q7: Can I use the point-slope formula to find the equation of a line that passes through three points?

A7: No, the point-slope formula is used to find the equation of a line that passes through two points. If you have three points, you can use the concept of the median to find the equation of the line.

Q8: What is the median of a line?

A8: The median of a line is the line that passes through the midpoint of the two given points. It is used to find the equation of a line that passes through three points.

Q9: How do I find the midpoint of two points?

A9: To find the midpoint of two points, you can use the formula: (x1 + x2) / 2, (y1 + y2) / 2.

Q10: Can I use the point-slope formula to find the equation of a line that is parallel to another line?

A10: No, the point-slope formula is used to find the equation of a line that passes through two given points. If you want to find the equation of a line that is parallel to another line, you can use the concept of parallel lines.