Finding The Equation Of A Line Slope Of 1 And Point (1, 5)

Introduction

Hey guys! Today, we're diving into a fundamental concept in mathematics: finding the equation of a line. Specifically, we're going to tackle a problem where we need to determine the equation of a line given its slope and a point it passes through. This is a classic problem that pops up in algebra and calculus, so understanding it is super important. We'll break down the steps, explain the underlying principles, and make sure you're comfortable solving similar problems on your own. Let's get started and make math a bit more fun!

The equation of a line is a crucial concept in mathematics, serving as the foundation for many advanced topics. Understanding how to find the equation of a line when given certain information, such as the slope and a point, is essential for students and anyone working with mathematical models. In this article, we will explore the process of determining the equation of a line that has a slope of 1 and passes through the point (1, 5). We'll walk through the concepts, formulas, and steps needed to solve this problem effectively. This knowledge not only helps in academic settings but also in real-world applications where linear relationships need to be modeled and understood.

Linear equations are the backbone of numerous mathematical and real-world applications. They describe relationships where a constant change in one variable results in a constant change in another. For instance, in physics, the equation of motion for an object moving at a constant velocity is a linear equation. In economics, linear equations can represent cost functions or supply-demand relationships. Therefore, mastering the art of finding the equation of a line is not just an academic exercise; it's a practical skill that can be applied in various fields. This article aims to provide a comprehensive guide, ensuring that readers grasp the underlying principles and can confidently apply them to solve related problems. We will use clear explanations, examples, and step-by-step instructions to make the process as straightforward as possible. Let's dive in and explore how to find the equation of the line with a slope of 1 that passes through the point (1, 5).

Understanding the Slope-Intercept Form

The slope-intercept form is a fundamental concept in understanding linear equations. The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept. The slope indicates the rate at which the line rises or falls, while the y-intercept is the point where the line crosses the y-axis. This form is incredibly useful because it provides a straightforward way to visualize and analyze the characteristics of a line. By knowing the slope and the y-intercept, we can easily graph the line and understand its behavior. The slope-intercept form is not just a formula; it’s a powerful tool for understanding linear relationships and making predictions based on linear models. Recognizing and utilizing this form is crucial for solving a wide range of mathematical problems involving linear equations.

In the equation y = mx + b, the slope m is the ratio of the change in y to the change in x, often described as "rise over run." A positive slope indicates that the line is increasing as you move from left to right, while a negative slope indicates that the line is decreasing. A slope of 0 represents a horizontal line, and an undefined slope represents a vertical line. The y-intercept b is the value of y when x is 0, which is the point where the line intersects the y-axis. Understanding these components is essential for manipulating and interpreting linear equations. The slope-intercept form allows us to quickly identify these key characteristics and use them to analyze the line's behavior. For instance, if we know the slope is 2 and the y-intercept is -3, we can immediately write the equation of the line as y = 2x - 3. This form makes it easy to graph the line and solve related problems.

The beauty of the slope-intercept form lies in its simplicity and versatility. It allows us to easily convert between different forms of linear equations and to quickly extract information about the line. For example, if we have the equation in standard form, Ax + By = C, we can rearrange it to the slope-intercept form to find the slope and y-intercept. This makes it easier to graph the line and understand its properties. Additionally, the slope-intercept form is widely used in various applications, from modeling physical phenomena to predicting trends in data. Its widespread use and practical applicability make it a fundamental concept in mathematics. By mastering the slope-intercept form, you gain a valuable tool for solving a wide array of problems and understanding the behavior of linear relationships.

Using the Point-Slope Form

Another crucial form for defining a line is the point-slope form. The point-slope form of a linear equation is given by y - y₁ = m(x - x₁), where m is the slope of the line, and (x₁, y₁) is a point that the line passes through. This form is particularly useful when we know a point on the line and the slope, but not the y-intercept. It allows us to construct the equation of the line directly from this information. The point-slope form is a powerful tool in algebra because it connects the geometric concept of a line to its algebraic representation. Understanding this form is essential for solving problems where we are given a point and a slope, and it provides a pathway to converting the equation into other forms, such as the slope-intercept form. By mastering the point-slope form, you gain a flexible and efficient method for working with linear equations.

The point-slope form, y - y₁ = m(x - x₁), is derived from the definition of slope. The slope m is the ratio of the change in y to the change in x, which can be expressed as m = (y - y₁) / (x - x₁). By multiplying both sides of this equation by (x - x₁), we arrive at the point-slope form. This derivation highlights the direct relationship between the slope and the coordinates of any point on the line. The point-slope form is especially advantageous when we need to find the equation of a line that passes through a specific point and has a particular slope. For instance, if we are given a slope of 3 and a point (2, 4), we can plug these values into the point-slope form to get y - 4 = 3(x - 2). This equation can then be simplified and converted into other forms, such as the slope-intercept form.

Using the point-slope form simplifies the process of finding the equation of a line, especially in scenarios where the y-intercept is not immediately apparent. This form is versatile and can be easily manipulated to solve various types of problems. For example, we can use it to find the equation of a line parallel or perpendicular to another line, given a point. The point-slope form is also a crucial stepping stone in understanding more advanced concepts in calculus and analytic geometry. Its flexibility and direct application make it an indispensable tool in mathematical problem-solving. By becoming proficient in using the point-slope form, you enhance your ability to analyze and solve linear equation problems efficiently and accurately.

Applying the Formulas to Our Problem

Now, let's apply these formulas to our specific problem. We are given that the line has a slope of 1 (m = 1) and passes through the point (1, 5). We need to find the equation of this line. We can use either the slope-intercept form or the point-slope form to solve this problem, but the point-slope form is often more direct when we have a point and a slope. We'll use both methods to demonstrate their application and show how they lead to the same result. This will help solidify your understanding and provide you with multiple approaches to solving similar problems. Let’s dive into the calculations and see how these formulas work in practice.

Using the point-slope form, y - y₁ = m(x - x₁), we substitute the given values. We have m = 1 and the point (x₁, y₁) = (1, 5). Plugging these values into the formula, we get y - 5 = 1(x - 1). This equation represents the line in point-slope form. To simplify it and convert it into slope-intercept form, we distribute the 1 on the right side, which gives us y - 5 = x - 1. Next, we add 5 to both sides of the equation to isolate y, resulting in y = x + 4. This is the equation of the line in slope-intercept form. The process demonstrates how straightforward it is to use the point-slope form to find the equation of a line when given a point and a slope.

Alternatively, we can use the slope-intercept form, y = mx + b, to solve the same problem. We know that m = 1, so the equation becomes y = 1x + b, or simply y = x + b. To find the value of b, the y-intercept, we can substitute the coordinates of the given point (1, 5) into the equation. This gives us 5 = 1 + b. Solving for b, we subtract 1 from both sides, resulting in b = 4. Thus, the equation of the line is y = x + 4, which is the same result we obtained using the point-slope form. This confirms that both methods are valid and lead to the same answer. Understanding both approaches allows you to choose the one that you find most convenient and efficient for a particular problem. By practicing both methods, you can build confidence and proficiency in finding the equations of lines.

Step-by-Step Solution

Let's break down the solution into a step-by-step process to ensure clarity.

1. Identify the given information: We have the slope m = 1 and the point (1, 5).
2. Choose a method: We can use either the point-slope form or the slope-intercept form. Let's start with the point-slope form, y - y₁ = m(x - x₁).
3. Substitute the values: Plug in m = 1, x₁ = 1, and y₁ = 5 into the point-slope form: y - 5 = 1(x - 1).
4. Simplify the equation: Distribute the 1 on the right side: y - 5 = x - 1.
5. Isolate y: Add 5 to both sides: y = x + 4.
6. Check your answer: Verify that the equation is in the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. In this case, m = 1 and b = 4.
7. Confirm the point: Substitute the given point (1, 5) into the equation y = x + 4: 5 = 1 + 4, which is true. This confirms that the line passes through the given point.

Following these steps systematically ensures that you arrive at the correct equation of the line. This process is not only applicable to this specific problem but also to a wide range of similar problems involving linear equations. Each step is designed to guide you through the logical progression of solving the problem, from identifying the given information to verifying the solution. This methodical approach helps prevent errors and ensures that you have a clear understanding of the solution. By practicing these steps, you will become more confident and efficient in solving linear equation problems.

These steps provide a clear and concise guide to solving the problem. By breaking down the process into manageable parts, we make it easier to understand and apply. This step-by-step approach is valuable not only for this specific problem but also for tackling other mathematical challenges. It encourages a systematic way of thinking and problem-solving, which is a crucial skill in mathematics and beyond. By mastering these steps, you are well-equipped to solve similar problems with confidence and accuracy.

Verifying the Solution

Verifying the solution is a crucial step in problem-solving. Verifying the solution ensures that the equation we found, y = x + 4, satisfies the given conditions: a slope of 1 and passing through the point (1, 5). There are several ways to verify the solution. We can visually inspect the equation to confirm that the slope is indeed 1. We can also substitute the coordinates of the given point into the equation to ensure that it holds true. Additionally, we can graph the line to visually confirm that it has the correct slope and passes through the specified point. This verification process is not just a formality; it's an essential step in ensuring the accuracy of our work and building confidence in our solution. Let's explore these verification methods in more detail.

First, let’s verify the slope. In the equation y = x + 4, the coefficient of x represents the slope. In this case, the coefficient is 1, which matches the given slope. This simple check confirms that the equation has the correct slope. Next, we will substitute the coordinates of the given point (1, 5) into the equation. If the equation holds true after the substitution, it confirms that the line passes through this point. Substituting x = 1 and y = 5 into y = x + 4, we get 5 = 1 + 4, which simplifies to 5 = 5. This is a true statement, confirming that the point (1, 5) lies on the line. These two checks provide a solid algebraic verification of our solution.

Finally, we can visualize the solution by graphing the line y = x + 4. Graphing provides a visual confirmation that the line has a slope of 1 and passes through the point (1, 5). When we plot the line, we can see that it rises one unit for every one unit it moves to the right, indicating a slope of 1. We can also see that the line intersects the y-axis at y = 4, which is the y-intercept in our equation. By plotting the point (1, 5), we can visually confirm that it lies on the line. This graphical verification complements the algebraic checks, providing a comprehensive confirmation of our solution. By employing these various verification methods, we can be confident that our equation y = x + 4 accurately represents the line with a slope of 1 that passes through the point (1, 5).

Conclusion

Wrapping things up, we've successfully found the equation of the line with a slope of 1 that passes through the point (1, 5). We've covered the importance of understanding linear equations, the slope-intercept and point-slope forms, and the step-by-step process to solve the problem. By using both the point-slope and slope-intercept forms, we arrived at the same equation, y = x + 4. We also emphasized the crucial step of verifying the solution to ensure accuracy. This comprehensive approach provides a solid foundation for tackling similar problems in the future. The skills and concepts learned here are fundamental in mathematics and have wide-ranging applications in various fields.

This problem highlights the significance of understanding different forms of linear equations and their applications. The ability to manipulate and solve linear equations is a fundamental skill in mathematics and is essential for many real-world applications. From physics and engineering to economics and computer science, linear equations are used to model and solve problems. By mastering the concepts presented in this article, you gain a valuable tool for your mathematical toolkit. The step-by-step approach we followed can be applied to a variety of problems, making you a more effective problem solver. Remember, practice is key to mastering these skills. The more you practice, the more confident and proficient you will become in working with linear equations.

In conclusion, the process of finding the equation of a line given its slope and a point is a fundamental skill in mathematics. By understanding the slope-intercept and point-slope forms, you can confidently tackle these types of problems. The step-by-step approach we've outlined, along with the verification methods, ensures accuracy and builds a strong foundation for more advanced mathematical concepts. Remember to practice and apply these skills to various problems to solidify your understanding. The equation y = x + 4 represents the line with a slope of 1 that passes through the point (1, 5), and we have successfully demonstrated how to arrive at this solution. Keep practicing, and you'll become a pro at solving linear equation problems!