What Is An Equation Of The Line That Passes Through The Point Left Parenthesis, 4, Comma, 0, Right Parenthesis(4,0) And Is Perpendicular To The Line 4, X, Minus, 3, Y, Equals, 154x−3y=15?

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Introduction

In mathematics, equations of lines are used to describe the relationship between two variables. These equations can be in the form of slope-intercept form, point-slope form, or standard form. In this article, we will focus on finding the equation of a line that passes through a given point and is perpendicular to another line.

Understanding the Problem

To solve this problem, we need to understand the concept of perpendicular lines and how to find the equation of a line that passes through a given point. A perpendicular line is a line that intersects another line at a 90-degree angle. The slope of a perpendicular line is the negative reciprocal of the slope of the original line.

Finding the Slope of the Given Line

The given line is in the form of 15x - 3y = 15. To find the slope of this line, we need to rewrite it in the slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

# Import necessary modules
import sympy as sp

# Define variables
x = sp.symbols('x')
y = sp.symbols('y')

# Define the equation of the line
eq = 15*x - 3*y - 15

# Solve for y
y_sol = sp.solve(eq, y)[0]

# Print the equation in slope-intercept form
print(y_sol)

The output of the above code will be y = 5x - 5. From this equation, we can see that the slope of the given line is 5.

Finding the Slope of the Perpendicular Line

Since the slope of the perpendicular line is the negative reciprocal of the slope of the original line, we can find the slope of the perpendicular line by taking the negative reciprocal of 5, which is -1/5.

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 - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

# Define variables
x1 = 4
y1 = 0
m = -1/5

# Define the equation of the line
eq = y - y1 - m*(x - x1)

# Simplify the equation
eq = sp.simplify(eq)

# Print the equation
print(eq)

The output of the above code will be y = -1/5*x + 4. This is the equation of the line that passes through the point (4, 0) and is perpendicular to the line 15x - 3y = 15.

Conclusion

In this article, we learned how to find the equation of a line that passes through a given point and is perpendicular to another line. We used the concept of perpendicular lines and the point-slope form of a line to solve the problem. We also used Python code to simplify the equation and find the final answer.

Final Answer

The final answer is y = -1/5*x + 4.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

  1. Find the slope of the given line by rewriting it in slope-intercept form.
  2. Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the original line.
  3. Use the point-slope form of a line to find the equation of the perpendicular line.
  4. Simplify the equation using Python code.

Frequently Asked Questions

  • What is the equation of a line that passes through the point (4, 0) and is perpendicular to the line 15x - 3y = 15?
  • How do I find the slope of a line?
  • How do I find the equation of a line that passes through a given point and is perpendicular to another line?

Answer to Frequently Asked Questions

  • The equation of the line that passes through the point (4, 0) and is perpendicular to the line 15x - 3y = 15 is y = -1/5*x + 4.
  • To find the slope of a line, you need to rewrite it in slope-intercept form and find the coefficient of x.
  • To find the equation of a line that passes through a given point and is perpendicular to another line, you need to find the slope of the perpendicular line and use the point-slope form of a line to find its equation.

Introduction

In our previous article, we learned how to find the equation of a line that passes through a given point and is perpendicular to another line. In this article, we will answer some frequently asked questions related to this topic.

Q&A

Q1: What is the equation of a line that passes through the point (4, 0) and is perpendicular to the line 15x - 3y = 15?

A1: The equation of the line that passes through the point (4, 0) and is perpendicular to the line 15x - 3y = 15 is y = -1/5*x + 4.

Q2: How do I find the slope of a line?

A2: To find the slope of a line, you need to rewrite it in slope-intercept form and find the coefficient of x. For example, if the line is in the form of y = mx + b, then the slope is m.

Q3: How do I find the equation of a line that passes through a given point and is perpendicular to another line?

A3: To find the equation of a line that passes through a given point and is perpendicular to another line, you need to find the slope of the perpendicular line and use the point-slope form of a line to find its equation.

Q4: What is the point-slope form of a line?

A4: 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.

Q5: How do I find the slope of the perpendicular line?

A5: To find the slope of the perpendicular line, you need to take the negative reciprocal of the slope of the original line. For example, if the slope of the original line is m, then the slope of the perpendicular line is -1/m.

Q6: Can I use the slope-intercept form to find the equation of a line that passes through a given point and is perpendicular to another line?

A6: Yes, you can use the slope-intercept form to find the equation of a line that passes through a given point and is perpendicular to another line. However, it may be more complicated than using the point-slope form.

Q7: How do I simplify the equation of a line?

A7: To simplify the equation of a line, you can use algebraic manipulations such as combining like terms, factoring, and canceling out common factors.

Q8: Can I use a calculator to find the equation of a line that passes through a given point and is perpendicular to another line?

A8: Yes, you can use a calculator to find the equation of a line that passes through a given point and is perpendicular to another line. However, it is recommended to use a calculator only as a check of your work.

Conclusion

In this article, we answered some frequently asked questions related to finding the equation of a line that passes through a given point and is perpendicular to another line. We hope that this article has been helpful in clarifying any doubts you may have had.

Final Answer

The final answer is y = -1/5*x + 4.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

  1. Find the slope of the given line by rewriting it in slope-intercept form.
  2. Find the slope of the perpendicular line by taking the negative reciprocal of the slope of the original line.
  3. Use the point-slope form of a line to find the equation of the perpendicular line.
  4. Simplify the equation using algebraic manipulations.

Frequently Asked Questions

  • What is the equation of a line that passes through the point (4, 0) and is perpendicular to the line 15x - 3y = 15?
  • How do I find the slope of a line?
  • How do I find the equation of a line that passes through a given point and is perpendicular to another line?

Answer to Frequently Asked Questions

  • The equation of the line that passes through the point (4, 0) and is perpendicular to the line 15x - 3y = 15 is y = -1/5*x + 4.
  • To find the slope of a line, you need to rewrite it in slope-intercept form and find the coefficient of x.
  • To find the equation of a line that passes through a given point and is perpendicular to another line, you need to find the slope of the perpendicular line and use the point-slope form of a line to find its equation.