Equation Of A Line Perpendicular To Another Line In Point-Slope Form

In mathematics, determining the equation of a line that satisfies specific conditions is a fundamental concept. This article delves into the process of finding the equation of a line that is perpendicular to a given line and passes through a specific point. We will focus on expressing the equation in point-slope form, a versatile representation that highlights both the slope and a point on the line. Let's explore the steps involved, using the example of finding the equation of a line perpendicular to y = (1/3)x + 7 that includes the point (2, 4).

Understanding Perpendicular Lines and Slopes

Before we dive into the calculations, it's crucial to grasp the relationship between perpendicular lines and their slopes. Perpendicular lines are lines that intersect at a right angle (90 degrees). A key property of perpendicular lines is that their slopes are negative reciprocals of each other. This means that if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. This is the fundamental principle that allows us to find the slope of the perpendicular line given the slope of the original line. Understanding this relationship is the cornerstone for solving problems involving perpendicular lines. We need to first identify the slope of the given line. In the equation y = (1/3)x + 7, the slope is 1/3. Therefore, the slope of the line perpendicular to it will be the negative reciprocal of 1/3, which is -3. This value will be crucial in formulating the equation of our new line.

Determining the Slope of the Perpendicular Line

The first crucial step in finding the equation of the perpendicular line is to determine its slope. As mentioned earlier, the slopes of perpendicular lines are negative reciprocals. The given line, y = (1/3)x + 7, is in slope-intercept form (y = mx + b), where m represents the slope. In this case, the slope of the given line is 1/3. To find the slope of a line perpendicular to this, we take the negative reciprocal of 1/3. The reciprocal of 1/3 is 3, and the negative of that is -3. Therefore, the slope of the line perpendicular to y = (1/3)x + 7 is -3. This value, -3, will be our m in the point-slope form equation. Knowing this slope is essential for constructing the equation of the line we're seeking. Without accurately determining the slope, the subsequent steps will lead to an incorrect equation. This concept is a cornerstone of coordinate geometry, and a firm grasp of it is crucial for solving a wide range of problems.

Point-Slope Form Explained

The point-slope form is a powerful and convenient way to represent the equation of a line. It directly incorporates a point on the line and the slope of the line. The general form of the point-slope equation is y - y₁ = m( x - x₁ ), where (x₁, y₁) represents a specific point on the line and m represents the slope of the line. This form is particularly useful when you know a point on the line and the slope, as it allows you to directly plug in these values to create the equation. The point-slope form provides a clear and intuitive way to understand the relationship between the slope, a point on the line, and the overall equation of the line. It bypasses the need to first calculate the y-intercept, which is required for the slope-intercept form. This makes it an efficient tool for finding the equation of a line when the necessary information is readily available in the form of a point and a slope. Understanding the structure and components of the point-slope form is critical for effectively using it in problem-solving.

Applying Point-Slope Form

Now that we have the slope of the perpendicular line (-3) and a point it passes through (2, 4), we can directly apply the point-slope form. The point-slope form equation is given by y - y₁ = m( x - x₁ ). We substitute the values we have: m = -3, x₁ = 2, and y₁ = 4. Plugging these values into the equation, we get: y - 4 = -3(x - 2). This is the equation of the line perpendicular to y = (1/3)x + 7 that includes the point (2, 4), expressed in point-slope form. This equation clearly shows the slope of the line (-3) and a specific point it passes through (2, 4). The point-slope form allows for a quick and easy representation of the line's equation based on these two pieces of information. Understanding how to apply the point-slope form is a valuable skill in linear algebra and coordinate geometry, enabling efficient solutions to various problems involving lines.

Substituting the Values

To use the point-slope form, we need to substitute the known values into the equation y - y₁ = m( x - x₁ ). We've already established that the slope (m) of our perpendicular line is -3, and the point (x₁, y₁) it passes through is (2, 4). Now we simply replace these variables in the equation with their corresponding values. Substituting y₁ with 4, m with -3, and x₁ with 2, we get: y - 4 = -3(x - 2). This substitution process is the core of using the point-slope form. It allows us to translate the geometrical information (slope and a point) into an algebraic equation representing the line. The resulting equation, y - 4 = -3(x - 2), is the answer we were seeking, expressed in the desired point-slope form. This demonstrates the power and simplicity of the point-slope form in finding the equation of a line when a point and a slope are known.

The Final Equation in Point-Slope Form

After substituting the values, we arrive at the final equation in point-slope form: y - 4 = -3(x - 2). This equation represents the line that is perpendicular to y = (1/3)x + 7 and passes through the point (2, 4). The equation clearly showcases the slope of the line, which is -3, and the point (2, 4) that lies on the line. This form is particularly useful because it directly displays the key information about the line – its slope and a point it passes through. The equation y - 4 = -3(x - 2) can be further manipulated into other forms, such as slope-intercept form (y = mx + b) or standard form (Ax + By = C), but the point-slope form is often the most convenient starting point when you have the slope and a point. It provides a concise and informative representation of the line's characteristics. Understanding and utilizing the point-slope form is a fundamental skill in linear algebra and analytical geometry.

Advantages of Point-Slope Form

The point-slope form offers several advantages when dealing with linear equations. First and foremost, it's incredibly straightforward to use when you know the slope of a line and a point that lies on it. Unlike the slope-intercept form, which requires you to calculate the y-intercept, the point-slope form allows you to directly plug in the known values of the slope and the point's coordinates. This makes it a faster and more efficient method in many situations. Additionally, the point-slope form provides a clear visual representation of the line's characteristics. The equation y - y₁ = m( x - x₁ ) immediately tells you the slope (m) and a point (x₁, y₁) on the line. This can be particularly helpful for understanding the line's orientation and position in the coordinate plane. Furthermore, the point-slope form serves as a strong foundation for converting to other forms of linear equations, such as slope-intercept form or standard form. It acts as a bridge, allowing you to easily manipulate the equation to fit different needs or preferences. In essence, the point-slope form is a versatile and powerful tool in linear algebra, offering a direct and informative way to represent linear equations.

In conclusion, we have successfully found the equation of the line perpendicular to y = (1/3)x + 7 that includes the point (2, 4), and expressed it in point-slope form: y - 4 = -3(x - 2). This process involved understanding the relationship between perpendicular lines and their slopes, applying the point-slope form equation, and substituting the given values to arrive at the final equation. This skill is fundamental in mathematics and has various applications in fields like physics, engineering, and computer graphics.