Finding The Equation Of A Perpendicular Line In Slope-Intercept Form
Finding the equation of a line that is perpendicular to a given line and passes through a specific point is a common problem in algebra. The slope-intercept form of a linear equation, which is y = mx + b, where m represents the slope and b represents the y-intercept, is a particularly useful way to express the equation of a line. In this article, we will explore the steps involved in determining the equation of a perpendicular line in slope-intercept form, using the given line y = -1/3x - 1/3 and the point (2, -1) as an example. Understanding how to manipulate linear equations and apply the concepts of slope and perpendicularity is crucial for success in various mathematical and real-world applications. Mastering these skills will enable you to solve a wide range of problems involving lines, slopes, and intercepts.
Understanding Slope and Perpendicular Lines
Before we dive into the calculations, let's make sure we understand some key concepts. The slope of a line describes its steepness and direction. A line with a positive slope rises from left to right, while a line with a negative slope falls from left to right. The slope is calculated as the change in y divided by the change in x between any two points on the line. Perpendicular lines, on the other hand, are lines that intersect at a right angle (90 degrees). The relationship between the slopes of perpendicular lines is very specific: they are negative reciprocals of each other. This means that if one line has a slope of m, the slope of a line perpendicular to it will be -1/m. This concept is essential for finding the equation of a perpendicular line.
In our case, the given line y = -1/3x - 1/3 has a slope of -1/3. To find the slope of a line perpendicular to this, we need to 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 we are looking for is 3. This understanding of slopes and perpendicularity forms the foundation for solving the problem. Without grasping these concepts, it would be impossible to determine the correct equation for the perpendicular line. The ability to identify and manipulate slopes is a fundamental skill in algebra and geometry, with applications in various fields, including physics, engineering, and computer graphics. Recognizing the inverse relationship between the slopes of perpendicular lines is key to solving geometric problems and understanding spatial relationships.
Finding the Slope of the Perpendicular Line
As mentioned earlier, the slopes of perpendicular lines are negative reciprocals of each other. The given line, y = -1/3x - 1/3, is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. By comparing the given equation to the slope-intercept form, we can easily identify the slope of the given line as -1/3. To find the slope of the line perpendicular to the given line, we need to calculate the negative reciprocal of -1/3. The reciprocal of a number is obtained by flipping the fraction, so the reciprocal of -1/3 is -3/1 or simply -3. The negative reciprocal is then the negative of -3, which is 3. Therefore, the slope of the line perpendicular to the given line is 3. This step is crucial because it provides us with the m value in the slope-intercept form (y = mx + b) for the perpendicular line. Without the correct slope, we cannot accurately determine the equation of the perpendicular line. This calculation highlights the importance of understanding the relationship between the slopes of perpendicular lines and how to manipulate fractions and negative signs. The ability to find the negative reciprocal is a key skill in solving problems involving perpendicular lines and geometric relationships.
Using the Point-Slope Form
Now that we have the slope (m = 3) of the perpendicular line, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. We are given the point (2, -1) that the perpendicular line passes through, so we can substitute x1 = 2 and y1 = -1 into the point-slope form. Plugging in these values and the slope m = 3, we get: y - (-1) = 3(x - 2). This equation represents the perpendicular line in point-slope form. The point-slope form is particularly useful when we know a point on the line and the slope, as it allows us to directly write the equation of the line without having to first find the y-intercept. The point-slope form is a versatile tool in linear algebra and is frequently used in various applications, including finding the equation of a tangent line to a curve in calculus. Understanding and being able to use the point-slope form is essential for solving a wide range of problems involving linear equations.
Converting to Slope-Intercept Form
While the point-slope form y - (-1) = 3(x - 2) is a valid equation for the perpendicular line, the question asks for the equation in slope-intercept form (y = mx + b). To convert from point-slope form to slope-intercept form, we need to simplify the equation and isolate y on one side. First, we can simplify the left side of the equation: y - (-1) becomes y + 1. Next, we distribute the 3 on the right side of the equation: 3(x - 2) becomes 3x - 6. So, our equation now looks like this: y + 1 = 3x - 6. To isolate y, we subtract 1 from both sides of the equation: y + 1 - 1 = 3x - 6 - 1. This simplifies to y = 3x - 7. Now the equation is in slope-intercept form, where the slope is 3 and the y-intercept is -7. This conversion process demonstrates the flexibility of linear equations and the ability to express the same relationship in different forms. Understanding how to convert between different forms of linear equations is crucial for solving various types of problems and for effectively communicating mathematical concepts.
The Final Equation
After converting the equation from point-slope form to slope-intercept form, we arrive at the final equation of the line perpendicular to y = -1/3x - 1/3 and passing through the point (2, -1). The equation is y = 3x - 7. This equation is in the desired slope-intercept form, where the slope is 3 and the y-intercept is -7. We can verify that this line is indeed perpendicular to the given line by checking that the slopes are negative reciprocals, which we confirmed earlier. We can also verify that the line passes through the point (2, -1) by substituting x = 2 into the equation and checking if we get y = -1: y = 3(2) - 7 = 6 - 7 = -1. This confirms that the point (2, -1) lies on the line y = 3x - 7. The ability to find the equation of a perpendicular line is a fundamental skill in algebra and geometry, with applications in various fields, including computer graphics, physics, and engineering. Understanding the relationship between slopes and intercepts and being able to manipulate linear equations are crucial for success in these areas.
In summary, we found the equation of the perpendicular line by first determining the slope of the perpendicular line (which is the negative reciprocal of the given line's slope), then using the point-slope form of a linear equation, and finally converting the equation to slope-intercept form. This process demonstrates a systematic approach to solving problems involving linear equations and geometric relationships. The final equation, y = 3x - 7, represents the line that meets the given conditions and provides a complete solution to the problem.
