Equation Of A Line Perpendicular To Another Line Passing Through A Point

In the realm of mathematics, specifically in coordinate geometry, finding the equation of a line that satisfies certain conditions is a fundamental skill. This article delves into the process of determining the equation of a line when given a point it passes through and the condition that it is perpendicular to another line. We will explore the underlying principles, the step-by-step method, and illustrate the process with a detailed example. The final equation will be expressed in either slope-intercept form or standard form, adhering to the convention of using integers or simplified fractions for numerical coefficients.

Understanding Perpendicular Lines and Slopes

Before we dive into the specific problem, it is crucial to understand the relationship between the slopes of perpendicular lines. Two lines are said to be perpendicular if they intersect at a right angle (90 degrees). A fundamental property of perpendicular lines is that the product of their slopes is -1, provided that neither line is vertical. In other words, if a line has a slope of m, then a line perpendicular to it will have a slope of -1/m. This is often referred to as the negative reciprocal of the slope.

To illustrate, consider 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 line perpendicular to it will have a slope of 4/3. This inverse relationship is the cornerstone of solving problems involving perpendicular lines. Grasping this concept of perpendicular lines and their slopes is paramount to finding the equation of a line that adheres to given conditions.

Moreover, it's important to remember the definition of slope itself. Slope, often denoted by the letter m, represents the steepness and direction of a line. It is calculated as the change in the vertical coordinate (rise) divided by the change in the horizontal coordinate (run) between any two points on the line. A positive slope indicates an upward slant from left to right, while a negative slope indicates a downward slant. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line. Understanding the slope concept ensures accurate calculation and is thus a crucial step to solving these types of problems. This thorough comprehension forms a solid base for the subsequent steps in determining the equation of the line.

The Slope-Intercept and Standard Forms of a Linear Equation

Linear equations, which represent straight lines on a coordinate plane, can be expressed in various forms, each offering unique insights into the line's properties. Two of the most common forms are the slope-intercept form and the standard form. Understanding these forms is essential for expressing the final answer in the desired format.

The slope-intercept form is given by the equation y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). This form is particularly useful for quickly identifying the slope and y-intercept of a line, making it easy to graph the line or compare it with other lines. When given a slope and a y-intercept, the equation can be directly written in this form.

The standard form of a linear equation is given by Ax + By = C, where A, B, and C are integers, and A is a non-negative integer. This form is often preferred when dealing with systems of linear equations or when it is necessary to eliminate fractions from the equation. Converting an equation from slope-intercept form to standard form involves algebraic manipulation, typically involving multiplying through by a common denominator to eliminate fractions and rearranging terms to fit the standard form structure. The slope-intercept and standard forms both effectively represent the same line, but the context of the problem often dictates which form is more appropriate for expressing the final answer. Mastering the conversion between these forms is a valuable skill in linear algebra. Thus, knowing how to interchange between these forms enables you to present your solution in the preferred format with ease.

Step-by-Step Method to Find the Equation

Now, let's outline the step-by-step method for finding the equation of a line that passes through a given point and is perpendicular to another line. This systematic approach will guide you through the process and ensure accuracy.

1. Determine the slope of the given line: If the equation of the given line is provided, identify its slope. If two points on the line are given, calculate the slope using the formula m = (y2 - y1) / (x2 - x1).
2. Calculate the slope of the perpendicular line: Find the negative reciprocal of the slope obtained in step 1. If the slope of the given line is m, the slope of the perpendicular line will be -1/m.
3. Use the point-slope form: Use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where m is the slope of the perpendicular line (calculated in step 2) and (x1, y1) is the given point that the line passes through. Substitute the values into the point-slope form.
4. Convert to slope-intercept or standard form: Simplify the equation obtained in step 3 and convert it to either slope-intercept form (y = mx + b) or standard form (Ax + By = C), depending on the instructions or the desired format. This may involve distributing, combining like terms, and rearranging the equation.
5. Verify the solution: As a final step, it's a good practice to verify your solution. You can substitute the coordinates of the given point into the equation you obtained to ensure it satisfies the equation. Additionally, you can check if the slopes of the two lines are negative reciprocals of each other. This step-by-step method facilitates the accurate and efficient solution of such problems. These steps provide a solid roadmap for solving these types of problems and ensure that you arrive at the correct answer consistently. Each step builds upon the previous one, ensuring a logical progression towards the solution.

Example: Finding the Equation of the Line

Let's apply this method to a specific example. Suppose we want to find the equation of the line that contains the point (-6, -8) and is perpendicular to a line with a slope of 2/3. We will follow the steps outlined in the previous section.

1. Determine the slope of the given line: The slope of the given line is 2/3.
2. Calculate the slope of the perpendicular line: The negative reciprocal of 2/3 is -3/2. Therefore, the slope of the line perpendicular to the given line is -3/2.
3. Use the point-slope form: Using the point-slope form y - y1 = m(x - x1), we substitute the point (-6, -8) and the slope -3/2: y - (-8) = (-3/2)(x - (-6)). This simplifies to y + 8 = (-3/2)(x + 6).
4. Convert to slope-intercept or standard form: Let's convert the equation to slope-intercept form (y = mx + b). Distribute the -3/2: y + 8 = (-3/2)x - 9. Subtract 8 from both sides: y = (-3/2)x - 17. This is the equation of the line in slope-intercept form. Now, let's convert it to standard form (Ax + By = C). Add (3/2)x to both sides: (3/2)x + y = -17. Multiply both sides by 2 to eliminate the fraction: 3x + 2y = -34. This is the equation of the line in standard form.
5. Verify the solution: Substitute the point (-6, -8) into the slope-intercept form: -8 = (-3/2)(-6) - 17. This simplifies to -8 = 9 - 17, which is true. The slopes 2/3 and -3/2 are also negative reciprocals of each other, confirming that the lines are perpendicular. This thorough verification ensures the accuracy of our solution. By following these steps meticulously, we arrive at the correct equation, expressed in both slope-intercept and standard forms. Thus, this illustrative example highlights the practical application of the theoretical concepts discussed earlier.

Conclusion

In conclusion, finding the equation of a line that satisfies specific conditions, such as passing through a given point and being perpendicular to another line, is a fundamental concept in mathematics. By understanding the relationship between the slopes of perpendicular lines, the point-slope form, and the slope-intercept and standard forms of linear equations, we can systematically solve these types of problems. The step-by-step method outlined in this article, along with the illustrative example, provides a clear and concise approach to finding the equation of a line under these conditions. Mastering this skill is essential for success in coordinate geometry and related mathematical fields. The ability to accurately determine these equations is thus a cornerstone of advanced mathematical problem-solving. Therefore, diligent practice and a thorough understanding of the concepts will pave the way for greater proficiency in mathematics.