Equation Of A Line Passing Through (1,5) With Slope Of 6
In the realm of coordinate geometry, understanding how to define a line given certain properties is a fundamental skill. One common scenario involves finding the equation of a line when you know a point it passes through and its slope. This article delves into the process of determining the equation of a line that passes through the point (1, 5) and has a slope of 6. We'll explore the underlying concepts, utilize the point-slope form of a linear equation, and present the final equation in both point-slope and slope-intercept forms. This comprehensive guide will not only provide the solution but also enhance your understanding of linear equations and their applications. Understanding the equation of a line is crucial in various fields, including physics, engineering, and economics, where linear relationships are frequently encountered. By mastering this concept, you'll be better equipped to model and analyze real-world phenomena. So, let's embark on this mathematical journey and unravel the equation of the line.
Understanding Slope and Points
Before diving into the specific problem, let's establish a firm understanding of the key concepts involved: slope and points. The slope of a line, often denoted by 'm', quantifies its steepness and direction. It represents the rate of change in the vertical direction (y-axis) for every unit change in the horizontal direction (x-axis). A positive slope indicates an upward slant, while a negative slope signifies a downward slant. A slope of zero represents a horizontal line, and an undefined slope corresponds to a vertical line. The slope is calculated using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two distinct points on the line. Points, on the other hand, are specific locations on the coordinate plane represented by ordered pairs (x, y). Each point uniquely identifies a position, and lines can be defined by the points they pass through. In our case, we are given the point (1, 5), which means the line passes through the location where x = 1 and y = 5. Understanding how slope and points interact is crucial for determining the equation of a line. The slope dictates the line's inclination, while the point anchors the line to a specific location on the plane. By combining this information, we can precisely define the line's path and express it mathematically. The concept of slope is not just confined to mathematics; it has practical applications in various fields. For instance, in construction, the slope of a roof is crucial for water drainage. In geography, the slope of a terrain determines the flow of rivers. Thus, mastering the concept of slope is not just an academic exercise but a valuable skill for real-world applications.
Point-Slope Form: A Powerful Tool
The point-slope form is a powerful tool for representing the equation of a line when we know a point on the line and its slope. This form provides a direct way to express the linear relationship without needing to calculate the y-intercept first. The general formula for the point-slope form is: y - y1 = m(x - x1), where (x1, y1) is a known point on the line and m is the slope. This equation essentially states that the change in y relative to the change in x is constant and equal to the slope. The beauty of the point-slope form lies in its simplicity and directness. It allows us to plug in the known values directly and obtain the equation of the line. In our specific problem, we are given the point (1, 5) and the slope 6. By substituting these values into the point-slope form, we can readily obtain the equation of the line. The point-slope form is particularly useful when dealing with problems where the y-intercept is not immediately apparent. It bypasses the need to calculate the y-intercept and provides a straightforward way to represent the line. Furthermore, the point-slope form can be easily converted to other forms, such as the slope-intercept form, if desired. This versatility makes it a valuable tool in solving various linear equation problems. The point-slope form is not just a mathematical abstraction; it has practical applications in various fields. For instance, in physics, it can be used to describe the motion of an object with constant velocity. In economics, it can be used to model linear cost functions. Thus, understanding the point-slope form is crucial for both theoretical understanding and practical problem-solving.
Applying Point-Slope Form to Our Problem
Now, let's apply the point-slope form to our specific problem. We are given that the line passes through the point (1, 5) and has a slope of 6. Using the point-slope form equation, y - y1 = m(x - x1), we substitute the given values. Here, (x1, y1) = (1, 5) and m = 6. Plugging these values into the equation, we get: y - 5 = 6(x - 1). This is the equation of the line in point-slope form. It directly represents the linear relationship between x and y based on the given point and slope. This form is particularly useful as it clearly displays the slope and a point on the line. To further understand the equation, we can simplify it or convert it to other forms, such as the slope-intercept form. The point-slope form provides a strong foundation for analyzing the line's properties and behavior. By substituting different values of x, we can find corresponding values of y, and vice versa. This allows us to trace the line's path and visualize its position on the coordinate plane. The equation y - 5 = 6(x - 1) encapsulates all the information about the line's inclination and position. It is a concise and powerful representation of the linear relationship. Understanding how to apply the point-slope form is crucial for solving various linear equation problems. It provides a systematic approach to finding the equation of a line given a point and a slope. This skill is not only essential in mathematics but also in various fields where linear relationships are prevalent.
Converting to Slope-Intercept Form
While the point-slope form is a valid representation of the line's equation, it's often useful to convert it to the slope-intercept form. The slope-intercept form is expressed as y = mx + b, where 'm' is the slope and 'b' is the y-intercept (the point where the line crosses the y-axis). This form is particularly convenient because it explicitly shows the slope and y-intercept, making it easy to visualize the line's behavior. To convert the equation from point-slope form (y - 5 = 6(x - 1)) to slope-intercept form, we need to isolate 'y' on one side of the equation. First, we distribute the 6 on the right side: y - 5 = 6x - 6. Next, we add 5 to both sides of the equation to isolate 'y': y = 6x - 6 + 5. Simplifying the equation, we get: y = 6x - 1. This is the equation of the line in slope-intercept form. We can see that the slope 'm' is 6, which matches the given information, and the y-intercept 'b' is -1. This means the line crosses the y-axis at the point (0, -1). The slope-intercept form provides a clear and concise representation of the line's properties. It allows us to quickly identify the slope and y-intercept, which are crucial for understanding the line's behavior. The slope-intercept form is widely used in various applications, such as graphing lines, analyzing linear relationships, and solving linear equations. Understanding how to convert between point-slope form and slope-intercept form is a valuable skill in mathematics. It allows us to choose the form that best suits the problem at hand and to easily extract information about the line's properties.
The Final Equations
In summary, we've successfully determined the equation of the line that passes through the point (1, 5) and has a slope of 6. We started by understanding the concepts of slope and points and then utilized the point-slope form to derive the equation. The point-slope form of the equation is: y - 5 = 6(x - 1). This equation directly represents the line based on the given point and slope. It's a concise and powerful representation that encapsulates all the necessary information. Furthermore, we converted the equation to slope-intercept form, which is: y = 6x - 1. This form explicitly shows the slope (6) and the y-intercept (-1), making it easy to visualize the line's behavior. Both forms of the equation are valid representations of the same line. The choice of which form to use often depends on the specific context or the information that needs to be emphasized. The point-slope form is particularly useful when the focus is on a specific point on the line, while the slope-intercept form is convenient for understanding the line's slope and y-intercept. Understanding how to derive and manipulate linear equations is a fundamental skill in mathematics. It allows us to model and analyze linear relationships, which are prevalent in various fields. By mastering this concept, you'll be better equipped to solve a wide range of problems and to apply mathematical principles to real-world situations. The journey of finding the equation of a line highlights the power of mathematical tools and the importance of understanding underlying concepts.
This article has provided a detailed explanation of how to find the equation of a line passing through a given point with a specific slope. We utilized the point-slope form and converted it to the slope-intercept form, demonstrating the versatility of linear equation representations. The key takeaway is the understanding of how slope and points define a line and how these properties can be expressed mathematically. Mastering this concept is crucial for various mathematical and real-world applications. Whether you're a student learning the fundamentals of algebra or a professional applying linear equations in your field, the principles discussed here will serve as a valuable foundation. Remember, the equation of a line is not just a mathematical formula; it's a powerful tool for modeling and understanding linear relationships. By grasping these concepts, you'll be able to tackle a wide range of problems and gain a deeper appreciation for the beauty and utility of mathematics. The ability to find the equation of a line is a stepping stone to more advanced mathematical concepts, such as systems of equations, linear transformations, and calculus. So, continue to explore the world of mathematics and discover the power of linear relationships.
