Equation Of A Line Find The Line With Slope 5 Through (4, 2)
In the realm of mathematics, linear equations hold a fundamental position, serving as the bedrock for more advanced concepts. Understanding how to manipulate and derive these equations is crucial for anyone venturing deeper into mathematical studies or practical applications. This article delves into the process of determining the equation of a line when provided with its slope and a point it passes through. Specifically, we will focus on finding the equation of a line with a slope of 5 that traverses the point (4, 2), expressing the answer in the slope-intercept form, which is a cornerstone of linear equation representation.
The slope-intercept form, denoted as y = mx + b, is 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 because it provides a clear and immediate understanding of the line's characteristics – its steepness (slope) and its position relative to the origin (y-intercept). Our mission is to manipulate the given information, the slope (m = 5) and a point (4, 2), to fit this form. The journey involves employing the point-slope form, a versatile tool in linear algebra, and transforming it into the desired slope-intercept form. By understanding this process, one gains not only the ability to solve similar problems but also a deeper appreciation for the elegance and interconnectedness of mathematical concepts.
The practical applications of finding the equation of a line are vast and varied. From predicting trends in data analysis to modeling physical phenomena in engineering, linear equations serve as essential tools. For instance, in economics, linear equations can model the relationship between supply and demand; in physics, they can describe the motion of an object under constant acceleration. Thus, mastering the techniques for deriving these equations is not just an academic exercise but a valuable skill with real-world implications. As we proceed, we will explore the underlying principles, step-by-step calculations, and the rationale behind each step, ensuring a comprehensive understanding of the process.
Understanding Slope-Intercept Form
To effectively find the equation of the line, it's imperative to have a solid grasp of the slope-intercept form: y = mx + b. As previously mentioned, 'm' represents the slope, which quantifies the line's steepness and direction. A slope of 5, in our case, indicates that for every one unit increase in x, the value of y increases by five units. The y-intercept, 'b', is the point where the line intersects the y-axis. It's the value of y when x is zero. Understanding these components is crucial because they uniquely define a line on the Cartesian plane.
The beauty of the slope-intercept form lies in its simplicity and intuitiveness. By simply looking at the equation, one can immediately discern the line's slope and its y-intercept. This makes it incredibly useful for graphing lines, comparing different lines, and understanding the relationship between the variables x and y. For instance, if we have two lines with different slopes, we know that one line is steeper than the other. Similarly, if they have different y-intercepts, we know they cross the y-axis at different points. This visual and conceptual clarity is why the slope-intercept form is so widely used in mathematics and its applications.
Furthermore, the slope-intercept form provides a direct link between algebra and geometry. Each equation represents a unique line, and each line can be described by a unique equation in slope-intercept form (unless the line is vertical, in which case it doesn't have a defined slope and cannot be represented in this form). This connection allows us to translate geometric problems into algebraic ones and vice versa, opening up a powerful arsenal of tools for problem-solving. For example, finding the intersection of two lines geometrically is equivalent to solving a system of two linear equations algebraically. This duality is a cornerstone of analytic geometry and underscores the importance of mastering the slope-intercept form.
The Point-Slope Form: A Crucial Tool
While the slope-intercept form is our ultimate destination, the point-slope form serves as a critical stepping stone in our journey. The point-slope form is expressed as y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point on the line. This form is particularly valuable when we have the slope and a point, as it allows us to directly plug in the values and obtain an equation of the line. In our case, we have m = 5 and the point (4, 2), making the point-slope form an ideal starting point.
The point-slope form is derived from the definition of slope itself. The slope, m, is defined as the change in y divided by the change in x, often written as (y2 - y1) / (x2 - x1). If we consider a general point on the line (x, y) and the given point (x1, y1), we can write the slope as (y - y1) / (x - x1). Setting this equal to the given slope, m, and multiplying both sides by (x - x1), we arrive at the point-slope form: y - y1 = m(x - x1). This derivation highlights the fundamental connection between the slope and the points on the line, making the point-slope form a natural choice for this type of problem.
The versatility of the point-slope form extends beyond just finding the equation of a line. It can also be used to check if a point lies on a line, to find the equation of a line parallel or perpendicular to a given line, or to solve various geometric problems involving lines. Its ability to directly incorporate a point and a slope makes it a powerful and flexible tool in linear algebra. Understanding and mastering the point-slope form is, therefore, essential for anyone seeking a deeper understanding of linear equations and their applications. It provides a bridge between the abstract concept of slope and the concrete reality of points on a line, allowing us to seamlessly navigate the world of linear relationships.
Applying Point-Slope Form to Our Problem
Now, let's put the point-slope form into action. We are given a slope of m = 5 and a point (4, 2). Plugging these values into the point-slope form, y - y1 = m(x - x1), we get y - 2 = 5(x - 4). This equation represents the line with the given slope that passes through the given point. However, it's not yet in the desired slope-intercept form. Our next step is to manipulate this equation algebraically to transform it into the familiar y = mx + b format.
The substitution step is straightforward but crucial. We replace m with 5, x1 with 4, and y1 with 2 in the point-slope formula. This direct application of the formula showcases its efficiency and ease of use. The resulting equation, y - 2 = 5(x - 4), encapsulates all the information we have about the line. It tells us the slope and a point through which the line passes. However, to fully understand the line's behavior and to easily compare it with other lines, we need to express it in the slope-intercept form.
The transition from point-slope form to slope-intercept form is a matter of algebraic manipulation. We need to distribute the 5 on the right side of the equation and then isolate y on the left side. This process involves applying the distributive property and adding a constant to both sides of the equation. These are fundamental algebraic operations that are essential for solving a wide range of mathematical problems. By mastering these techniques, we gain the ability to manipulate equations and transform them into different forms, allowing us to extract the information we need and to solve problems efficiently.
Converting to Slope-Intercept Form
To convert the equation y - 2 = 5(x - 4) into slope-intercept form, the initial step involves applying the distributive property. This means multiplying the 5 by both terms inside the parentheses: 5 * x and 5 * -4. This yields the equation y - 2 = 5x - 20. The distributive property is a fundamental concept in algebra, allowing us to simplify expressions and equations by removing parentheses. It is a cornerstone of algebraic manipulation and is crucial for solving various types of equations.
Next, we need to isolate y on the left side of the equation. To do this, we add 2 to both sides of the equation. This operation maintains the equality of the equation while moving the constant term to the right side. Adding 2 to both sides of y - 2 = 5x - 20 gives us y = 5x - 18. This is the final step in the conversion process, and it results in the equation being in slope-intercept form.
The resulting equation, y = 5x - 18, is now in slope-intercept form (y = mx + b). We can clearly see that the slope, m, is 5, which was given initially. The y-intercept, b, is -18, which means the line intersects the y-axis at the point (0, -18). This equation provides a complete description of the line, allowing us to easily graph it, compare it with other lines, and understand its relationship with the coordinate plane. The process of converting from point-slope form to slope-intercept form demonstrates the power of algebraic manipulation and its ability to transform equations into more useful and informative forms.
The Final Equation and Its Significance
After successfully converting the equation, we arrive at the final answer in slope-intercept form: y = 5x - 18. This equation represents the line with a slope of 5 that passes through the point (4, 2). The slope of 5 indicates the line's steepness – for every one unit increase in x, y increases by five units. The y-intercept of -18 tells us that the line crosses the y-axis at the point (0, -18). This information allows us to fully visualize and understand the line's position and orientation on the coordinate plane.
The significance of this equation extends beyond just representing a line. It serves as a model for various real-world phenomena. For instance, if we consider x as time and y as distance, this equation could represent the motion of an object moving at a constant velocity of 5 units per time unit, starting from a position 18 units behind the origin. Similarly, in economics, this equation could represent a cost function, where x is the number of units produced, y is the total cost, 5 is the variable cost per unit, and -18 is a fixed cost (or a subsidy, in this case). These examples illustrate the broad applicability of linear equations in modeling and understanding various aspects of the world around us.
Furthermore, this exercise highlights the power of mathematical tools and techniques in solving problems. By understanding the slope-intercept form, the point-slope form, and the process of converting between them, we can confidently tackle similar problems and gain a deeper appreciation for the elegance and interconnectedness of mathematical concepts. The ability to find the equation of a line given its slope and a point is a fundamental skill in mathematics and a stepping stone to more advanced topics in algebra, calculus, and beyond.
In conclusion, finding the equation of a line with a given slope that passes through a specific point is a fundamental skill in mathematics, with broad applications across various fields. By understanding the slope-intercept form, the point-slope form, and the process of converting between them, we can effectively solve such problems. In our specific example, we successfully determined that the equation of the line with a slope of 5 that passes through the point (4, 2) is y = 5x - 18. This equation provides a complete description of the line, allowing us to visualize its position and orientation on the coordinate plane and to use it as a model for real-world phenomena.
The process of solving this problem reinforces the importance of understanding mathematical concepts and techniques. The slope-intercept form and the point-slope form are not just formulas to be memorized; they are powerful tools that allow us to understand and manipulate linear relationships. The algebraic manipulations involved in converting between these forms demonstrate the power of mathematical reasoning and problem-solving skills. By mastering these concepts and techniques, we gain the ability to tackle more complex mathematical problems and to apply mathematical thinking to various aspects of our lives.
Moreover, the ability to find the equation of a line is a stepping stone to more advanced topics in mathematics. Linear equations are the foundation for systems of equations, linear inequalities, and linear programming, which are essential tools in various fields such as economics, engineering, and computer science. Understanding linear equations is also crucial for calculus, where the concept of the derivative is based on the slope of a tangent line. Therefore, mastering the techniques for finding the equation of a line is an investment in one's future mathematical endeavors and a key to unlocking more advanced mathematical concepts.
Linear Equations, Slope-Intercept Form, Point-Slope Form, Equation of a Line, Slope, Y-Intercept, Algebraic Manipulation, Mathematical Problem-Solving, Cartesian Plane, Real-World Applications, Mathematics, Equation
