Determine The Equation Of A Perpendicular Line A Comprehensive Guide

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In the realm of coordinate geometry, the concept of perpendicular lines holds significant importance. Understanding how to determine the equation of a line perpendicular to a given line is a fundamental skill. This article delves into a step-by-step approach to solve this type of problem, using the specific example of finding the equation of a line passing through the point (-3, -7) and perpendicular to the line 4x + 5y = 10. This comprehensive guide will not only provide the solution but also illuminate the underlying principles and concepts, making it a valuable resource for students and enthusiasts alike.

Understanding Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). A crucial property of perpendicular lines is the relationship between their slopes. The slopes of two perpendicular lines are negative reciprocals of each other. This means that if a line has a slope of m, a line perpendicular to it will have a slope of -1/m. This inverse relationship is the key to solving problems involving perpendicular lines.

The Significance of Slope

The slope of a line is a measure of its steepness and direction. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. The slope is often represented by the letter m. Understanding the concept of slope is crucial for grasping the relationship between perpendicular lines. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. A steeper line has a larger absolute value of the slope.

The Negative Reciprocal Relationship

The negative reciprocal relationship between the slopes of perpendicular lines is a fundamental geometric principle. If we have a line with a slope of 2, a line perpendicular to it will have a slope of -1/2. Similarly, if a line has a slope of -3/4, a perpendicular line will have a slope of 4/3. This relationship allows us to determine the slope of a perpendicular line if we know the slope of the original line. This concept is not just a mathematical abstraction; it has practical applications in various fields, including architecture, engineering, and computer graphics.

Step-by-Step Solution

To determine the equation of the line through the point (-3, -7) that is perpendicular to the line 4x + 5y = 10, we will follow a step-by-step approach:

  1. Find the slope of the given line: We need to rewrite the equation 4x + 5y = 10 in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

    • Subtract 4x from both sides: 5y = -4x + 10
    • Divide both sides by 5: y = (-4/5)x + 2
    • Therefore, the slope of the given line is -4/5.

  2. Determine the slope of the perpendicular line: The slope of a line perpendicular to the given line is the negative reciprocal of -4/5, which is 5/4.

    • The negative reciprocal of a fraction is obtained by flipping the fraction and changing its sign. In this case, we flip -4/5 to get -5/4, and then change the sign to get 5/4.

  3. Use the point-slope form to find the equation: The point-slope form of a linear equation is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

    • We have the point (-3, -7) and the slope 5/4. Plugging these values into the point-slope form, we get:
      • y - (-7) = (5/4)(x - (-3))
      • y + 7 = (5/4)(x + 3)

  4. Simplify the equation to slope-intercept or standard form: We can simplify the equation to either slope-intercept form (y = mx + b) or standard form (Ax + By = C). Let's convert it to standard form.

    • Multiply both sides by 4 to eliminate the fraction: 4(y + 7) = 5(x + 3)
    • Distribute: 4y + 28 = 5x + 15
    • Rearrange the terms to get the standard form: 5x - 4y = 13

Therefore, the equation of the line through the point (-3, -7) that is perpendicular to the line 4x + 5y = 10 is 5x - 4y = 13.

In-Depth Explanation of Each Step

To fully grasp the solution, let's delve into each step with greater detail:

1. Finding the Slope of the Given Line

The first step involves transforming the given equation, 4x + 5y = 10, into slope-intercept form, which is y = mx + b. This form is crucial because it explicitly reveals the slope (m) and the y-intercept (b) of the line. The slope-intercept form provides a clear and intuitive way to understand the line's behavior. The coefficient of x in this form directly represents the slope, which is the rate of change of y with respect to x.

The Importance of Slope-Intercept Form

The slope-intercept form is a fundamental concept in linear algebra and coordinate geometry. It provides a direct way to visualize and analyze linear equations. The slope (m) tells us how much the y-value changes for every unit change in the x-value. A positive slope indicates an increasing line, while a negative slope indicates a decreasing line. The y-intercept (b) is the point where the line crosses the y-axis, which is the value of y when x is zero.

Converting to Slope-Intercept Form

The process of converting the given equation to slope-intercept form involves isolating y on one side of the equation. This is achieved through a series of algebraic manipulations, ensuring that the equation remains balanced. First, we subtract 4x from both sides of the equation, which isolates the term containing y. Then, we divide both sides by 5 to obtain y by itself. This process transforms the equation into the desired form, revealing the slope and y-intercept.

2. Determining the Slope of the Perpendicular Line

Once we have the slope of the given line (-4/5), we can determine the slope of the perpendicular line using the principle of negative reciprocals. The negative reciprocal of a number is obtained by flipping the fraction and changing its sign. This relationship is a direct consequence of the geometric properties of perpendicular lines.

The Geometric Basis of the Negative Reciprocal Relationship

The negative reciprocal relationship between the slopes of perpendicular lines stems from the fact that perpendicular lines intersect at a right angle. The slopes of the lines determine the angles they make with the x-axis. When two lines are perpendicular, the product of their slopes is -1. This relationship can be derived using trigonometric principles and the properties of right triangles.

Calculating the Negative Reciprocal

To find the negative reciprocal of -4/5, we first flip the fraction to get -5/4, and then change the sign to get 5/4. This new slope, 5/4, represents the slope of any line that is perpendicular to the original line. It's important to understand that this relationship is not arbitrary; it's a fundamental property of perpendicular lines that has deep roots in geometry.

3. Using the Point-Slope Form

The point-slope form of a linear equation, y - y1 = m(x - x1), is a powerful tool for finding the equation of a line when we know a point on the line (x1, y1) and the slope (m). This form is particularly useful when we have a point and a slope but need to express the equation in a more general form, such as slope-intercept or standard form.

Understanding the Point-Slope Form

The point-slope form is derived from the definition of slope. The slope (m) is the ratio of the change in y to the change in x between any two points on the line. If we have a specific point (x1, y1) on the line and another arbitrary point (x, y), the slope can be expressed as (y - y1) / (x - x1). Setting this equal to the slope m and rearranging the terms gives us the point-slope form.

Applying the Point-Slope Form

In this case, we have the point (-3, -7) and the slope 5/4. Plugging these values into the point-slope form, we get y - (-7) = (5/4)(x - (-3)). This equation represents the line that passes through the point (-3, -7) and has a slope of 5/4. The point-slope form provides a concise and direct way to express this relationship.

4. Simplifying the Equation

After obtaining the equation in point-slope form, we simplify it to either slope-intercept form (y = mx + b) or standard form (Ax + By = C). The standard form is often preferred because it provides a more general representation of the line and avoids fractions. Converting to standard form involves rearranging the terms so that the equation is in the form Ax + By = C, where A, B, and C are integers.

Converting to Standard Form

The process of converting the equation to standard form involves several algebraic steps. First, we multiply both sides of the equation by 4 to eliminate the fraction. Then, we distribute the constants on both sides. Finally, we rearrange the terms to get the equation in the form 5x - 4y = 13. This equation is in standard form and represents the line that is perpendicular to the given line and passes through the given point.

Alternative Solution Methods

While the step-by-step method outlined above is a standard approach, there are alternative methods to solve this problem. One such method involves directly using the standard form of a linear equation and applying the properties of perpendicular lines.

Using the Standard Form Directly

The standard form of a linear equation is Ax + By = C. If we have a line in this form, the slope is -A/B. For a line perpendicular to this, the slope would be B/A. Using this, we can directly form the equation of the perpendicular line.

Steps for Alternative Solution

  1. Identify A and B: In the equation 4x + 5y = 10, A = 4 and B = 5.
  2. Determine the coefficients of the perpendicular line: The coefficients of the perpendicular line will be 5x - 4y = C.
  3. Substitute the point (-3, -7) to find C: 5(-3) - 4(-7) = C, which gives C = 13.
  4. The equation of the perpendicular line is 5x - 4y = 13.

This alternative method provides a more direct approach and can be particularly useful for those who are comfortable working with the standard form of a linear equation.

Common Mistakes and How to Avoid Them

Solving problems involving perpendicular lines requires careful attention to detail. Here are some common mistakes and how to avoid them:

  1. Incorrectly Calculating the Negative Reciprocal: A common mistake is to forget to both flip the fraction and change the sign. Always remember to do both to find the negative reciprocal.
  2. Using the Wrong Slope: It's crucial to use the slope of the perpendicular line, not the slope of the original line, when finding the equation of the perpendicular line.
  3. Algebraic Errors: Mistakes in simplifying the equation can lead to incorrect results. Double-check each step of the algebraic manipulation to ensure accuracy.
  4. Misunderstanding the Point-Slope Form: Ensure you correctly substitute the point and slope into the point-slope form to avoid errors.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving these types of problems.

Real-World Applications

The concepts of perpendicular lines and their equations are not just theoretical exercises; they have numerous real-world applications in various fields:

  1. Architecture: Architects use perpendicular lines to design buildings and structures, ensuring stability and aesthetic appeal.
  2. Engineering: Engineers rely on perpendicular lines in structural design, road construction, and other projects where right angles are critical.
  3. Computer Graphics: Perpendicular lines are used in computer graphics to create 3D models and animations.
  4. Navigation: Navigational systems use perpendicular lines to determine directions and plot courses.
  5. Physics: In physics, perpendicular vectors are used to analyze forces and motion.

Understanding the properties of perpendicular lines is essential for anyone working in these fields.

Conclusion

Determining the equation of a line perpendicular to a given line is a fundamental skill in coordinate geometry. By understanding the relationship between the slopes of perpendicular lines and applying the point-slope form, we can effectively solve these types of problems. This article has provided a comprehensive guide, including a step-by-step solution, an in-depth explanation of each step, alternative solution methods, common mistakes to avoid, and real-world applications. With practice and a solid understanding of the underlying principles, you can confidently tackle any problem involving perpendicular lines.

By mastering these concepts, you gain a valuable tool for solving a wide range of problems in mathematics and beyond. The ability to work with perpendicular lines is not just a mathematical skill; it's a way of thinking that can be applied to various aspects of life and work.