What Is The Point-slope Equation Of The Line With Slope -10 That Passes Through The Point (1, 4)?

In the realm of coordinate geometry, understanding the equations of lines is foundational. Among the various forms of linear equations, the point-slope form holds a special place due to its intuitive nature and ease of use. This form allows us to define a line using a single point on the line and its slope, making it a powerful tool for solving a variety of problems. The point-slope equation provides a straightforward method to express the equation of a line when you know a point on the line and its slope. The point-slope form of a linear equation is given by: y - y₁ = m(x - x₁), where (x₁, y₁) is a known point on the line and m is the slope of the line. This form is particularly useful because it directly incorporates the geometric properties of a line—its steepness (slope) and a specific location (point). Understanding this equation is crucial for various applications in mathematics, physics, engineering, and computer graphics. For instance, in physics, it can describe the motion of an object along a straight path, while in computer graphics, it's used to draw lines on a screen. In mathematics, this form is often used as a stepping stone to finding other forms of linear equations, such as the slope-intercept form or the standard form. The elegance of the point-slope form lies in its direct representation of the line's characteristics. It's a tangible link between the abstract equation and the visual line on a coordinate plane. By understanding how to use this form, you gain a deeper insight into the nature of linear relationships, which are fundamental to many mathematical and real-world phenomena. The point-slope equation is not just a formula; it's a window into the behavior of lines and their role in describing the world around us. This understanding is essential for anyone delving into mathematical problem-solving and practical applications. Therefore, grasping the essence of the point-slope form is an investment in your mathematical toolkit, opening doors to more complex concepts and applications.

The derivation of the point-slope equation is a beautiful illustration of how mathematical formulas arise from fundamental principles. It starts with the definition of the slope of a line, which is the ratio of the change in the y-coordinate to the change in the x-coordinate between any two points on the line. Let's consider a line with a slope m and a known point (x₁, y₁) on this line. Now, let's take any other arbitrary point (x, y) on the same line. The slope m can be expressed as: m = (y - y₁) / (x - x₁). This equation simply states that the slope is the "rise" (change in y) divided by the "run" (change in x). To arrive at the point-slope form, we multiply both sides of this equation by (x - x₁). This gives us: y - y₁ = m(x - x₁). This is the point-slope equation. It elegantly captures the relationship between the slope, a known point, and any other point on the line. The beauty of this derivation is its simplicity and clarity. It shows how a fundamental concept like the slope can be used to create a powerful equation that describes a line. By rearranging the definition of the slope, we've unlocked a way to express the equation of a line using minimal information—a single point and the slope. This equation is not just a formula to be memorized; it's a logical consequence of the definition of the slope. This understanding is crucial because it allows you to not only use the formula but also to appreciate its origins and limitations. Understanding the derivation also makes it easier to remember the formula. Instead of rote memorization, you can reconstruct the equation from the basic principles of slope. This approach fosters a deeper understanding of mathematics and enhances your problem-solving skills. The derivation of the point-slope equation is a testament to the power of mathematical reasoning. It's a reminder that mathematical formulas are not arbitrary constructs but are logical outcomes of fundamental definitions and principles. By grasping the derivation, you gain a more profound understanding of the equation and its applications.

To determine the point-slope equation of a line, you need two key pieces of information: a point on the line and the slope of the line. Once you have these, the process is straightforward. The point-slope form is given by: y - y₁ = m(x - x₁). Where: * (x₁, y₁) is the known point on the line. * m is the slope of the line. The process involves substituting the given values into the equation and simplifying. This straightforward substitution is a key advantage of the point-slope form. It allows you to directly translate the geometric properties of the line into an algebraic equation. For example, if you are given a point (2, 3) and a slope of -1, you would substitute these values into the formula as follows: * x₁ = 2 * y₁ = 3 * m = -1 This gives you the equation: y - 3 = -1(x - 2). This equation represents the line that passes through the point (2, 3) and has a slope of -1. It's important to note that this is not the only way to express the equation of this line. You could also convert it to slope-intercept form (y = mx + b) or standard form (Ax + By = C). However, the point-slope form is particularly useful when you are given a point and a slope because it allows you to write the equation directly without having to solve for any additional parameters. The point-slope equation is a powerful tool for solving problems in coordinate geometry. It allows you to express the equation of a line in a concise and meaningful way. By mastering this form, you gain a valuable skill for tackling a wide range of mathematical problems. The beauty of the point-slope form lies in its directness and ease of use. It allows you to quickly capture the essential characteristics of a line in a single equation. This is why it is a fundamental concept in mathematics and is widely used in various applications.

In this specific problem, we are asked to find the point-slope equation of a line with a slope of -10 that passes through the point (1, 4). We are given: * Slope (m) = -10 * Point (x₁, y₁) = (1, 4) To find the equation, we substitute these values into the point-slope equation: y - y₁ = m(x - x₁). Substituting the given values, we get: y - 4 = -10(x - 1). This equation represents the line that passes through the point (1, 4) and has a slope of -10. Now, let's examine the given options to see which one matches our result: * A. y + 4 = -10(x + 1) * B. y - 4 = 10(x - 1) * C. y + 4 = 10(x + 1) * D. y - 4 = -10(x - 1) Comparing our derived equation with the options, we can see that option D, y - 4 = -10(x - 1), matches exactly. Therefore, option D is the correct answer. This problem demonstrates the direct application of the point-slope form. By simply substituting the given values into the formula, we can easily find the equation of the line. This is a testament to the power and simplicity of the point-slope form. The point-slope equation is a versatile tool that can be used to solve a wide range of problems in coordinate geometry. It is particularly useful when you are given a point and a slope because it allows you to write the equation directly without having to solve for any additional parameters. In this case, the problem was solved by directly applying the formula and comparing the result with the given options. This approach highlights the importance of understanding the formula and its applications. The solution to this problem is a clear demonstration of the effectiveness of the point-slope form. It showcases how a simple substitution can lead to a correct answer, making it a valuable tool in your mathematical arsenal.

The point-slope equation is not just a theoretical construct; it has numerous practical applications across various fields. Its importance stems from its ability to describe linear relationships in a concise and meaningful way. In mathematics, it serves as a building block for understanding more complex concepts, such as linear transformations and systems of equations. It is also used extensively in calculus for finding tangent lines to curves. In physics, the point-slope equation can be used to describe motion with constant velocity. For example, if you know the initial position and velocity of an object, you can use the point-slope form to write an equation that describes its position at any time. This is a fundamental concept in kinematics, the study of motion. In engineering, the point-slope form is used in various applications, such as designing bridges and buildings. Engineers need to understand the forces acting on a structure, and linear equations are often used to model these forces. The point-slope form can be used to write equations that describe the relationship between these forces. In computer graphics, the point-slope form is used to draw lines on a screen. Computer graphics systems use coordinate systems to represent images, and lines are fundamental building blocks of these images. The point-slope form provides a way to define lines using a point and a slope, which is essential for drawing them on a screen. The applications of the point-slope equation are vast and varied. It is a fundamental tool in many fields, from mathematics and physics to engineering and computer graphics. Its versatility and ease of use make it an indispensable part of any problem-solver's toolkit. Understanding the point-slope form is not just about memorizing a formula; it's about gaining a deeper understanding of linear relationships and their applications in the real world. This understanding empowers you to tackle a wide range of problems and make meaningful contributions in various fields. The significance of the point-slope equation extends beyond the classroom. It is a powerful tool that can be used to solve real-world problems and advance our understanding of the world around us.

In conclusion, the point-slope equation is a fundamental concept in coordinate geometry with far-reaching applications. Its ability to define a line using a single point and its slope makes it an invaluable tool for solving a variety of problems. From its derivation to its practical applications, the point-slope form provides a clear and concise way to express linear relationships. Understanding this equation is crucial for anyone delving into mathematics, physics, engineering, or computer graphics. It's not just a formula to be memorized, but a powerful tool that can unlock a deeper understanding of the world around us. The point-slope equation is a testament to the elegance and power of mathematics. It is a reminder that simple concepts can have profound implications and that understanding the fundamentals is key to success in any field. By mastering the point-slope form, you equip yourself with a valuable skill that will serve you well in your academic and professional pursuits. The importance of the point-slope equation cannot be overstated. It is a cornerstone of mathematical thinking and a gateway to more advanced concepts. So, embrace the point-slope form, understand its derivation, and explore its applications. You will find that it is a powerful ally in your journey of learning and discovery.