Solving 1/3x - 5 + 171 = X A Step-by-Step Guide
Linear equations are fundamental in mathematics and are used extensively in various fields, from physics and engineering to economics and computer science. Understanding how to solve linear equations is a crucial skill for anyone pursuing studies or careers in these areas. In this comprehensive guide, we will delve into solving the linear equation 1/3x - 5 + 171 = x, providing a step-by-step explanation to ensure clarity and comprehension. Our goal is to break down the process into manageable steps, making it accessible even to those who are new to algebra. Mastering the art of solving linear equations like this one not only builds a strong foundation in algebra but also enhances problem-solving abilities that are transferable to many other domains. This article will serve as a valuable resource for students, educators, and anyone looking to refresh their algebra skills. We will cover each step in detail, emphasizing the underlying principles and techniques involved. By the end of this guide, you should feel confident in your ability to tackle similar linear equations independently. The journey of understanding mathematics is often built on mastering the basics, and solving linear equations is a cornerstone of that journey. So, let's embark on this mathematical exploration together and demystify the process of solving 1/3x - 5 + 171 = x.
Understanding Linear Equations
Before diving into the specific equation, it's crucial to grasp the concept of linear equations. Linear equations are algebraic expressions where the highest power of the variable (in this case, 'x') is 1. They represent a straight line when graphed on a coordinate plane, hence the name 'linear'. These equations are characterized by having a constant rate of change, meaning that for every unit increase in 'x', the value of the expression changes by a constant amount. This consistent relationship makes linear equations incredibly useful for modeling real-world phenomena. The general form of a linear equation is ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable. Solving a linear equation means finding the value of 'x' that makes the equation true. This involves isolating 'x' on one side of the equation by performing the same operations on both sides to maintain balance. Understanding this fundamental principle is key to solving any linear equation. The beauty of linear equations lies in their simplicity and predictability. Unlike more complex equations, linear equations have straightforward solutions that can be found using a few basic algebraic techniques. Recognizing the linear nature of an equation is the first step towards solving it effectively. The equation 1/3x - 5 + 171 = x fits this definition, making it a prime example of a linear equation that we can solve using algebraic methods. Let's now move on to the process of solving this specific equation, breaking it down into clear, manageable steps.
Step 1: Simplify the Equation
The first step in solving any equation, including our linear equation 1/3x - 5 + 171 = x, is to simplify both sides of the equation as much as possible. Simplification involves combining like terms and reducing the equation to its most basic form. This not only makes the equation easier to work with but also reduces the chances of making errors in subsequent steps. In our equation, we can identify constant terms on the left side: -5 and +171. These are like terms because they do not involve the variable 'x'. To simplify, we add these constants together: -5 + 171 = 166. This means we can rewrite the equation as 1/3x + 166 = x. This simplification step is crucial as it condenses the equation, making the next steps more straightforward. Simplifying equations is a fundamental technique in algebra and is applicable to a wide range of problems. By reducing the equation to its simplest form, we can clearly see the relationship between the variable and the constants, which is essential for isolating the variable and finding its value. Remember, the goal of simplification is to make the equation as manageable as possible without changing its fundamental meaning. The simplified equation, 1/3x + 166 = x, is now ready for the next steps in the solution process. We have successfully combined the constant terms, paving the way for further algebraic manipulations to isolate 'x'.
Step 2: Isolate the Variable Terms
After simplifying the equation, the next crucial step is to isolate the variable terms on one side of the equation. Isolating the variable means getting all terms involving 'x' on one side and all constant terms on the other side. This is a fundamental strategy in solving equations because it brings us closer to finding the value of 'x'. In our equation, 1/3x + 166 = x, we have 'x' terms on both sides. To isolate them, we need to move the 1/3x term from the left side to the right side. We can achieve this by subtracting 1/3x from both sides of the equation. This maintains the equality and allows us to consolidate the 'x' terms. Subtracting 1/3x from both sides gives us: 166 = x - 1/3x. Now, we have all the 'x' terms on the right side, which is a significant step towards solving the equation. The process of isolating the variable is a cornerstone of algebraic manipulation. It allows us to separate the unknown ('x') from the known constants, making it possible to determine its value. This step often involves adding or subtracting terms from both sides of the equation, ensuring that the balance is maintained. Isolating the variable terms is a strategic move that sets the stage for the final steps in solving the equation. With the variable terms now on one side, we can proceed to simplify them further and ultimately find the value of 'x'.
Step 3: Combine Like Terms with the Variable
Now that we have isolated the variable terms on one side of the equation, the next step is to combine these like terms. Combining like terms involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. In our equation, 166 = x - 1/3x, we have two terms involving 'x': 'x' and '-1/3x'. To combine these, we need to treat 'x' as 1x and subtract 1/3x from it. This requires finding a common denominator to perform the subtraction. We can rewrite 'x' as 3/3x, which has the same denominator as 1/3x. Now we can subtract: 3/3x - 1/3x = 2/3x. This simplifies the equation to 166 = 2/3x. This step is crucial because it consolidates the variable terms into a single term, making the equation much easier to solve. Combining like terms is a fundamental skill in algebra and is essential for simplifying expressions and equations. It allows us to reduce the complexity of the equation and bring it closer to a solution. The ability to identify and combine like terms is a key component of algebraic proficiency. By combining the 'x' terms, we have transformed the equation into a more manageable form, ready for the final step of isolating 'x'. The equation 166 = 2/3x is now poised for us to solve for the value of 'x'.
Step 4: Solve for x
After simplifying and combining like terms, the final step in solving the linear equation 1/3x - 5 + 171 = x is to isolate 'x' and find its value. Solving for x involves performing the necessary operations to get 'x' by itself on one side of the equation. Our equation is now 166 = 2/3x. To isolate 'x', we need to undo the multiplication by 2/3. The inverse operation of multiplying by a fraction is multiplying by its reciprocal. The reciprocal of 2/3 is 3/2. So, we multiply both sides of the equation by 3/2 to isolate 'x'. This gives us (3/2) * 166 = (3/2) * (2/3x). On the left side, (3/2) * 166 simplifies to 249. On the right side, (3/2) * (2/3x) simplifies to x because the fractions cancel each other out. Therefore, we have 249 = x. This means the solution to the equation is x = 249. We have successfully solved the linear equation by isolating 'x' and finding its value. Solving for the variable is the ultimate goal in solving any equation. It involves using inverse operations to undo the operations that are being performed on the variable. This step requires careful attention to maintain the balance of the equation by performing the same operation on both sides. The solution x = 249 is the value that makes the original equation true. By substituting this value back into the original equation, we can verify its correctness. The ability to solve for variables is a fundamental skill in mathematics and is essential for various applications. With x = 249, we have completed the process of solving the linear equation 1/3x - 5 + 171 = x.
Verification of the Solution
After solving an equation, it's always a good practice to verify the solution. Verification involves substituting the value you found for 'x' back into the original equation to ensure it makes the equation true. This step is crucial for catching any potential errors in your calculations and confirming the accuracy of your solution. Our original equation was 1/3x - 5 + 171 = x, and we found that x = 249. To verify, we substitute 249 for 'x' in the equation: 1/3(249) - 5 + 171 = 249. First, we calculate 1/3 of 249, which is 83. So the equation becomes: 83 - 5 + 171 = 249. Next, we simplify the left side of the equation: 83 - 5 = 78, and 78 + 171 = 249. Thus, the equation becomes: 249 = 249. Since both sides of the equation are equal, our solution x = 249 is correct. Verification is a vital step in the problem-solving process. It provides confidence in your answer and ensures that you have not made any mistakes along the way. This practice is particularly important in algebra, where even a small error can lead to an incorrect solution. By verifying the solution, we confirm that the value of 'x' we found satisfies the original equation, making it the correct answer. The verification step completes the solution process and provides assurance in the accuracy of our result. With the solution verified, we can confidently conclude that x = 249 is the solution to the linear equation 1/3x - 5 + 171 = x.
Conclusion
In this comprehensive guide, we have walked through the process of solving the linear equation 1/3x - 5 + 171 = x, step by step. From simplifying the equation to isolating the variable and verifying the solution, we have covered each aspect in detail. Mastering the techniques for solving linear equations is a fundamental skill in mathematics, and this example serves as a solid foundation for tackling more complex problems. We began by understanding the concept of linear equations and their general form. Then, we systematically simplified the equation by combining like terms. Next, we isolated the variable terms on one side of the equation and combined them to make the equation easier to solve. Finally, we solved for 'x' by using inverse operations and verified our solution by substituting it back into the original equation. The solution we found was x = 249, which satisfies the equation. This step-by-step approach not only helps in solving this specific equation but also provides a framework for solving other linear equations. The ability to solve equations is a crucial skill that extends beyond mathematics and is applicable in various fields. By understanding the underlying principles and practicing these techniques, you can build confidence in your problem-solving abilities. Linear equations are a building block for more advanced mathematical concepts, and a strong grasp of them is essential for further studies in mathematics and related fields. This guide aims to provide a clear and accessible explanation of the solution process, empowering you to tackle similar problems with ease and confidence.
