Finding the Slope of a Given Line

Oct 13, 2025 by thesweetdesigns.com 23 views

**Finding the Equation of a Perpendicular Line** =====================================================

Introduction

In mathematics, finding the equation of a line that is perpendicular to a given line is a common problem. This involves understanding the concept of slope and how it relates to perpendicular lines. In this article, we will explore how to find the equation of a line that contains a given point and is perpendicular to a given line.

Understanding Slope

The slope of a line is a measure of how steep it is. It is calculated by dividing the vertical change (rise) by the horizontal change (run). The slope of a line can be represented by the letter 'm'. For example, if a line has a slope of 2, it means that for every 1 unit of horizontal change, the line rises 2 units.

Finding the Slope of a Given Line

To find the slope of a given line, we can use the slope-intercept form of a line, which is y = mx + b, where m is the slope and b is the y-intercept. In the given problem, the line is y = -2x + 8. To find the slope, we can compare this equation to the slope-intercept form.

y = -2x + 8
m = -2

Finding the Slope of a Perpendicular Line

The slope of a perpendicular line is the negative reciprocal of the slope of the given line. This means that if the slope of the given line is m, the slope of the perpendicular line is -1/m.

m = -2
m_perpendicular = -1/m
m_perpendicular = -1/(-2)
m_perpendicular = 1/2

Finding the Equation of a Perpendicular Line

Now that we have the slope of the perpendicular line, we can use the point-slope form of a line to find its equation. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

y - (-2) = (1/2)(x - 6)
y + 2 = (1/2)x - 3
y = (1/2)x - 5

Conclusion

In this article, we have learned how to find the equation of a line that contains a given point and is perpendicular to a given line. We have also learned how to find the slope of a given line and the slope of a perpendicular line. By following these steps, we can find the equation of a perpendicular line.

Q&A

Q: What is the slope of a perpendicular line?

A: The slope of a perpendicular line is the negative reciprocal of the slope of the given line.

Q: How do I find the equation of a perpendicular line?

A: To find the equation of a perpendicular line, you can use the point-slope form of a line and substitute the slope of the perpendicular line and a point on the line.

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

A: 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.

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

A: To find the slope of a given line, you can use the slope-intercept form of a line and compare it to the given equation.

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

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

Q: How do I find the equation of a line that contains a given point and is perpendicular to a given line?

A: To find the equation of a line that contains a given point and is perpendicular to a given line, you can use the point-slope form of a line and substitute the slope of the perpendicular line and the given point.

Q: What is the negative reciprocal of a number?

A: The negative reciprocal of a number is the number with the opposite sign and the reciprocal of the number.

Q: How do I find the negative reciprocal of a number?

A: To find the negative reciprocal of a number, you can divide 1 by the number and change the sign of the result.

Q: What is the reciprocal of a number?

A: The reciprocal of a number is 1 divided by the number.

Q: How do I find the reciprocal of a number?

A: To find the reciprocal of a number, you can divide 1 by the number.

Q: What is the equation of a line that contains the point (6, -2) and is perpendicular to the line y = -2x + 8?

A: The equation of a line that contains the point (6, -2) and is perpendicular to the line y = -2x + 8 is y = (1/2)x - 5.