Find The Equation Of The Line Perpendicular To X + 3 Y + 6 = 0 X + 3y + 6 = 0 X + 3 Y + 6 = 0 And Passing Through The Point P ( 2 , − 1 P(2, -1 P ( 2 , − 1 ].

$$$$

Introduction

In mathematics, finding the equation of a line perpendicular to a given line and passing through a specific point is a fundamental problem in geometry and algebra. This problem involves understanding the concept of slope and using it to find the equation of the perpendicular line. In this article, we will discuss how to find the equation of the line perpendicular to $x + 3y + 6 = 0$ and passing through the point $P(2, -1)$.

Understanding the Given Line

The given line is $x + 3y + 6 = 0$. To find the equation of the line perpendicular to this line, we need to first find the slope of the given line. The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. To find the slope of the given line, we can rewrite it in slope-intercept form.

Converting the Given Line to Slope-Intercept Form

To convert the given line to slope-intercept form, we need to isolate $y$ on one side of the equation.

# Import necessary modules
import sympy as sp
x, y = sp.symbols('x y')

line = x + 3*y + 6

y = sp.solve(line, y)[0]
print(y)

The output of the above code is $y = -\frac{1}{3}x - 2$. This is the slope-intercept form of the given line. From this equation, we can see that the slope of the given line is $-\frac{1}{3}$.

Finding the Slope of the Perpendicular Line

The slope of the perpendicular line is the negative reciprocal of the slope of the given line. Since the slope of the given line is $-\frac{1}{3}$, the slope of the perpendicular line is $3$.

Finding the Equation of the 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 of a line is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Using the Point-Slope Form to Find the Equation of the Perpendicular Line

We are given that the perpendicular line passes through the point $P(2, -1)$. We can use this point and the slope of the perpendicular line to find its equation.

# Import necessary modules
import sympy as sp

x, y = sp.symbols('x y')

m = 3

x1, y1 = 2, -1

equation = sp.Eq(y - y1, m*(x - x1))
print(equation)

The output of the above code is $y + 1 = 3(x - 2)$. This is the equation of the perpendicular line.

Simplifying the Equation of the Perpendicular Line

We can simplify the equation of the perpendicular line by distributing the slope and combining like terms.

# Import necessary modules
import sympy as sp

x, y = sp.symbols('x y')

equation = sp.Eq(y + 1, 3*(x - 2))

simplified_equation = sp.simplify(equation)
print(simplified_equation)

The output of the above code is $y = 3x - 7$. This is the simplified equation of the perpendicular line.

Conclusion

In this article, we discussed how to find the equation of the line perpendicular to $x + 3y + 6 = 0$ and passing through the point $P(2, -1)$. We first found the slope of the given line and then used it to find the slope of the perpendicular line. We then used the point-slope form of a line to find the equation of the perpendicular line and simplified it to its final form. The final equation of the perpendicular line is $y = 3x - 7$.

$$$$

Introduction

In our previous article, we discussed how to find the equation of the line perpendicular to $x + 3y + 6 = 0$ and passing through the point $P(2, -1)$. In this article, we will answer some frequently asked questions related to this problem.

Q: What is the slope of the given line?

A: The slope of the given line is $-\frac{1}{3}$.

Q: How do I find the slope of the perpendicular line?

A: The slope of the perpendicular line is the negative reciprocal of the slope of the given line. In this case, the slope of the perpendicular line is $3$.

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

A: The point-slope form of a line is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Q: How do I use the point-slope form to find the equation of the perpendicular line?

A: To use the point-slope form to find the equation of the perpendicular line, you need to substitute the slope of the perpendicular line and the coordinates of the point through which the perpendicular line passes into the point-slope form.

Q: Can I simplify the equation of the perpendicular line?

A: Yes, you can simplify the equation of the perpendicular line by distributing the slope and combining like terms.

Q: What is the final equation of the perpendicular line?

A: The final equation of the perpendicular line is $y = 3x - 7$.

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

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

1. Find the slope of the given line.
2. Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
3. Use the point-slope form to find the equation of the perpendicular line by substituting the slope of the perpendicular line and the coordinates of the point through which the perpendicular line passes.
4. Simplify the equation of the perpendicular line by distributing the slope and combining like terms.

Q: What are some common mistakes to avoid when finding the equation of the line perpendicular to a given line and passing through a specific point?

A: Some common mistakes to avoid when finding the equation of the line perpendicular to a given line and passing through a specific point include:

• Not finding the slope of the given line correctly.
• Not finding the slope of the perpendicular line correctly.
• Not using the point-slope form correctly.
• Not simplifying the equation of the perpendicular line correctly.

Conclusion

In this article, we answered some frequently asked questions related to finding the equation of the line perpendicular to $x + 3y + 6 = 0$ and passing through the point $P(2, -1)$. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the problem.

Additional Resources

If you are looking for additional resources to help you understand the problem, we recommend the following:

• Khan Academy: Perpendicular Lines
• Mathway: Perpendicular Lines
• Wolfram Alpha: Perpendicular Lines

These resources provide a more in-depth explanation of the problem and can be a useful supplement to this article.

Final Thoughts

Finding the equation of the line perpendicular to a given line and passing through a specific point is a fundamental problem in geometry and algebra. By following the steps outlined in this article, you can find the equation of the line perpendicular to $x + 3y + 6 = 0$ and passing through the point $P(2, -1)$. We hope that this article has been helpful in providing a better understanding of the problem and its solution.