Given The Equation Of Line \[$ L_1 \$\]:$\[ y - 6 = 3x + 3 + 6 \\ y = 3x + 9 \\]Find The Equation Of \[$ L_2 \$\], Which Is Perpendicular To Line \[$ L_1 \$\] And Passes Through The Point \[$(-1, 6)\$\], In The

Introduction

In mathematics, the concept of perpendicular lines is a fundamental idea in geometry and algebra. Two lines are said to be perpendicular if they intersect at a right angle, which is 90 degrees. In this article, we will explore how to find the equation of a line that is perpendicular to a given line and passes through a specific point.

The Given Line

The equation of the given line, denoted as $l_1$, is:

$$y - 6 = 3x + 3 + 6$$

Simplifying the equation, we get:

$$y = 3x + 9$$

This is the equation of a line with a slope of 3 and a y-intercept of 9.

The Perpendicular Line

To find the equation of the perpendicular line, denoted as $l_2$, we need to follow these steps:

1. Find the Slope of the Perpendicular Line: The slope of the perpendicular line is the negative reciprocal of the slope of the given line. In this case, the slope of the given line is 3, so the slope of the perpendicular line is -1/3.
2. Use the Point-Slope Form: The point-slope form of a line is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

In this case, the point is $(-1, 6)$ and the slope is -1/3. Plugging these values into the point-slope form, we get:

$$y - 6 = -\frac{1}{3}(x + 1)$$

Simplifying the Equation

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

$$3(y - 6) = -1(x + 1)$$

Expanding and simplifying, we get:

$$3y - 18 = -x - 1$$

Adding 18 to both sides and adding x to both sides, we get:

$$3y = -x + 17$$

Dividing both sides by 3, we get:

$$y = -\frac{1}{3}x + \frac{17}{3}$$

Conclusion

In this article, we found the equation of a line that is perpendicular to a given line and passes through a specific point. We used the point-slope form of a line and the negative reciprocal of the slope of the given line to find the equation of the perpendicular line. The final equation of the perpendicular line is:

$$y = -\frac{1}{3}x + \frac{17}{3}$$

This equation represents a line with a slope of -1/3 and a y-intercept of 17/3.

Key Takeaways

• The slope of the perpendicular line is the negative reciprocal of the slope of the given line.
• The point-slope form of a line is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

• To find the equation of a perpendicular line, we can use the point-slope form and the negative reciprocal of the slope of the given line.

Practice Problems

1. Find the equation of a line that is perpendicular to the line $y = 2x + 1$ and passes through the point $(0, 3)$.
2. Find the equation of a line that is perpendicular to the line $y = -x + 2$ and passes through the point $(1, 4)$.

Solutions

1. The slope of the given line is 2, so the slope of the perpendicular line is -1/2. Using the point-slope form, we get:

$$y - 3 = -\frac{1}{2}(x - 0)$$

Simplifying, we get:

$$y = -\frac{1}{2}x + 3$$

2. The slope of the given line is -1, so the slope of the perpendicular line is 1. Using the point-slope form, we get:

$$y - 4 = 1(x - 1)$$

Simplifying, we get:

$$y = x + 3$$

Conclusion

In this article, we found the equation of a line that is perpendicular to a given line and passes through a specific point. We used the point-slope form of a line and the negative reciprocal of the slope of the given line to find the equation of the perpendicular line. The final equation of the perpendicular line is:

$$y = -\frac{1}{3}x + \frac{17}{3}$$

This equation represents a line with a slope of -1/3 and a y-intercept of 17/3.

References

• [1] Khan Academy. (n.d.). Perpendicular Lines. Retrieved from https://www.khanacademy.org/math/geometry/geometry-lines/geometry-perpendicular-lines/v/perpendicular-lines
• [2] Math Open Reference. (n.d.). Perpendicular Lines. Retrieved from https://www.mathopenref.com/lineperpendicular.html

Glossary

• Perpendicular Lines: Two lines that intersect at a right angle, which is 90 degrees.
• Slope: A measure of how steep a line is. It is calculated as the ratio of the vertical change to the horizontal change.
• Point-Slope Form: A way of writing the equation of a line using the slope and a point on the line.
  Perpendicular Lines Q&A ==========================

Introduction

In our previous article, we explored how to find the equation of a line that is perpendicular to a given line and passes through a specific point. In this article, we will answer some frequently asked questions about perpendicular lines.

Q: What is the difference between perpendicular lines and parallel lines?

A: Perpendicular lines are lines that intersect at a right angle, which is 90 degrees. Parallel lines, on the other hand, are lines that never intersect and are always the same distance apart.

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

A: To find the equation of a line that is perpendicular to a given line, you need to follow these steps:

1. Find the Slope of the Given Line: The slope of the given line is the ratio of the vertical change to the horizontal change.
2. Find the Negative Reciprocal of the Slope: The negative reciprocal of the slope is the slope of the perpendicular line.
3. Use the Point-Slope Form: The point-slope form of a line is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

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

A: The point-slope form of a line is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

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

A: To find the equation of a line that is perpendicular to a given line and passes through a specific point, you need to follow these steps:

1. Find the Slope of the Given Line: The slope of the given line is the ratio of the vertical change to the horizontal change.
2. Find the Negative Reciprocal of the Slope: The negative reciprocal of the slope is the slope of the perpendicular line.
3. Use the Point-Slope Form: The point-slope form of a line is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Q: What is the difference between the slope of a line and the slope of a perpendicular line?

A: The slope of a line is the ratio of the vertical change to the horizontal change. The slope of a perpendicular line is the negative reciprocal of the slope of the original line.

Q: How do I find the equation of a line that is perpendicular to a given line and passes through a specific point using the slope-intercept form?

A: To find the equation of a line that is perpendicular to a given line and passes through a specific point using the slope-intercept form, you need to follow these steps:

1. Find the Slope of the Given Line: The slope of the given line is the ratio of the vertical change to the horizontal change.
2. Find the Negative Reciprocal of the Slope: The negative reciprocal of the slope is the slope of the perpendicular line.
3. Use the 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.

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

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

$$y = mx + b$$

where $m$ is the slope and $b$ is the y-intercept. The point-slope form of a line is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Conclusion

In this article, we answered some frequently asked questions about perpendicular lines. We covered topics such as the difference between perpendicular lines and parallel lines, how to find the equation of a line that is perpendicular to a given line, and the difference between the slope-intercept form and the point-slope form of a line.

References

• [1] Khan Academy. (n.d.). Perpendicular Lines. Retrieved from https://www.khanacademy.org/math/geometry/geometry-lines/geometry-perpendicular-lines/v/perpendicular-lines
• [2] Math Open Reference. (n.d.). Perpendicular Lines. Retrieved from https://www.mathopenref.com/lineperpendicular.html

Glossary

• Perpendicular Lines: Two lines that intersect at a right angle, which is 90 degrees.
• Slope: A measure of how steep a line is. It is calculated as the ratio of the vertical change to the horizontal change.
• Point-Slope Form: A way of writing the equation of a line using the slope and a point on the line.
• Slope-Intercept Form: A way of writing the equation of a line using the slope and the y-intercept.