Use The Given Conditions To Write An Equation For The Line In Point-slope Form And In Slope-intercept Form.Conditions:- The Line Passes Through The Point { (2, -6)$} . − T H E L I N E I S P E R P E N D I C U L A R T O T H E L I N E W I T H T H E E Q U A T I O N \[ .- The Line Is Perpendicular To The Line With The Equation \[ . − T H E L In E I S P Er P E N D I C U L A R T O T H E L In E W I T H T H Ee Q U A T I O N \[ Y =

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Introduction

In mathematics, the equation of a line can be represented in various forms, including point-slope form and slope-intercept form. The point-slope form is given by the equation {y - y_1 = m(x - x_1)$}$, where {m$}$ is the slope of the line and {(x_1, y_1)$}$ is a point on the line. The slope-intercept form, on the other hand, is given by the equation {y = mx + b$}$, where {m$}$ is the slope and {b$}$ is the y-intercept. In this article, we will use the given conditions to write an equation for the line in point-slope form and in slope-intercept form.

Given Conditions

The line passes through the point {(2, -6)$}$ and is perpendicular to the line with the equation {y = \frac{1}{2}x + 3$}$.

Step 1: Find the Slope of the Perpendicular Line

To find the slope of the perpendicular line, we need to find the negative reciprocal of the slope of the given line. The slope of the given line is {\frac{1}{2}$], so the slope of the perpendicular line is [-\frac{2}{1} = -2\$}.

Step 2: Write the Equation of the Perpendicular Line in Point-Slope Form

Using the point-slope form, we can write the equation of the perpendicular line as {y - (-6) = -2(x - 2)$}$. Simplifying this equation, we get {y + 6 = -2x + 4$}$.

Step 3: Write the Equation of the Perpendicular Line in Slope-Intercept Form

To write the equation of the perpendicular line in slope-intercept form, we need to isolate the variable {y$. Subtracting [6} from both sides of the equation, we get {y = -2x - 2$}$.

Conclusion

In this article, we used the given conditions to write an equation for the line in point-slope form and in slope-intercept form. We found the slope of the perpendicular line, wrote the equation of the perpendicular line in point-slope form, and then converted it to slope-intercept form. The final equation of the perpendicular line in slope-intercept form is {y = -2x - 2$}$.

Key Takeaways

  • The point-slope form of a line is given by the equation {y - y_1 = m(x - x_1)$].
  • The slope-intercept form of a line is given by the equation [y = mx + b\$}.
  • To find the slope of a perpendicular line, we need to find the negative reciprocal of the slope of the given line.
  • To write the equation of a line in slope-intercept form, we need to isolate the variable {y$].

Further Reading

For more information on the equation of a line, including point-slope form and slope-intercept form, please refer to the following resources:

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

A: The point-slope form of a line is given by the equation [y−y1=m(x−x1)$],where\[y - y_1 = m(x - x_1)\$], where \[m$}$ is the slope of the line and {(x_1, y_1)$}$ is a point on the line.

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 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 = \frac{y_2 - y_1}{x_2 - x_1}$], where [(x_1, y_1)\$} and {(x_2, y_2)$}$ are two points on the line.

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

A: To find the y-intercept of a line, you can use the formula {b = y - mx$}$, where {m$}$ is the slope and {(x, y)$}$ is a point on the line.

Q: What is the relationship between the slope and the y-intercept of a line?

A: The slope and the y-intercept of a line are related by the equation {y = mx + b$}$, where {m$}$ is the slope and {b$}$ is the y-intercept.

Q: How do I write the equation of a line in point-slope form?

A: To write the equation of a line in point-slope form, you can use the formula {y - y_1 = m(x - x_1)$], where [m\$} is the slope and {(x_1, y_1)$}$ is a point on the line.

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

A: To write the equation of a line in slope-intercept form, you can use the formula {y = mx + b$}$, where {m$}$ is the slope and {b$}$ is the y-intercept.

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

A: The point-slope form of a line is given by the equation {y - y_1 = m(x - x_1)$], while the slope-intercept form of a line is given by the equation [y = mx + b\$}. The point-slope form is useful when you know the slope and a point on the line, while the slope-intercept form is useful when you know the slope and the y-intercept.

Q: How do I find the equation of a line that passes through two points?

A: To find the equation of a line that passes through two points, you can use the formula {y - y_1 = m(x - x_1)$], where [m\$} is the slope and {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are the two points on the line.

Q: How do I find the equation of a line that is perpendicular to another line?

A: To find the equation of a line that is perpendicular to another line, you can use the formula {m = -\frac{1}{m}$], where [m\$} is the slope of the original line. Then, you can use the formula {y - y_1 = m(x - x_1)$], where [m\$} is the slope of the perpendicular line and {(x_1, y_1)$}$ is a point on the line.

Conclusion

In this article, we have answered some frequently asked questions about the equation of a line. We have discussed the point-slope form and the slope-intercept form of a line, and we have provided formulas for finding the slope and the y-intercept of a line. We have also discussed how to write the equation of a line in point-slope form and in slope-intercept form, and we have provided examples of how to find the equation of a line that passes through two points and how to find the equation of a line that is perpendicular to another line.