Solving 10x - 4.5 + 3x = 12x - 1.1 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the backbone of numerous mathematical concepts and real-world applications. These equations involve variables raised to the power of one, and the goal is to isolate the variable and determine its value. This article delves into the intricacies of solving the linear equation 10x - 4.5 + 3x = 12x - 1.1, providing a step-by-step guide to arriving at the correct solution. We will not only solve the equation but also explore the underlying principles and techniques that make solving linear equations a breeze. Understanding these concepts is crucial for anyone venturing into algebra and beyond. Whether you're a student grappling with homework or simply someone looking to sharpen your math skills, this guide will equip you with the knowledge and confidence to tackle linear equations effectively. So, let's embark on this mathematical journey together, unravel the mysteries of linear equations, and discover the elegant solutions that lie within.

At the heart of algebra lies the linear equation, a mathematical statement that asserts the equality between two expressions. These expressions involve variables, typically represented by letters like 'x', raised to the power of one. The beauty of linear equations lies in their simplicity and predictability. They can be visualized as straight lines on a graph, hence the name 'linear'. Understanding the structure of a linear equation is the first step towards mastering the art of solving them. A typical linear equation takes the form ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable we aim to find. The constants 'a' and 'b' represent the slope and y-intercept of the line, respectively. The goal of solving a linear equation is to isolate the variable 'x' on one side of the equation, thereby determining its value. This involves performing a series of algebraic manipulations while maintaining the equality of both sides. These manipulations include adding or subtracting the same value from both sides, multiplying or dividing both sides by the same non-zero value, and combining like terms. By carefully applying these techniques, we can systematically unravel the equation and reveal the solution. Linear equations are not just abstract mathematical concepts; they are powerful tools for modeling real-world scenarios. From calculating the distance traveled by a car to determining the optimal price for a product, linear equations provide a framework for understanding and solving a wide range of problems. Their ubiquity in various fields, including physics, engineering, economics, and computer science, underscores the importance of mastering the art of solving them. So, as we delve deeper into the techniques for solving linear equations, remember that we are not just manipulating symbols; we are unlocking the power to describe and understand the world around us.

Now, let's put our understanding of linear equations into practice by solving the given equation: 10x - 4.5 + 3x = 12x - 1.1. This equation, like any linear equation, can be solved by systematically isolating the variable 'x'. The first step in this process is to simplify both sides of the equation by combining like terms. On the left side, we have two terms involving 'x': 10x and 3x. Adding these together gives us 13x. So, the left side simplifies to 13x - 4.5. On the right side, we have 12x - 1.1, which is already in its simplest form. Our equation now looks like this: 13x - 4.5 = 12x - 1.1. The next step is to move all the terms involving 'x' to one side of the equation and all the constant terms to the other side. To do this, we can subtract 12x from both sides of the equation. This eliminates the 'x' term from the right side and leaves us with: 13x - 12x - 4.5 = 12x - 12x - 1.1, which simplifies to x - 4.5 = -1.1. Now, we need to isolate 'x' completely. To do this, we can add 4.5 to both sides of the equation. This eliminates the constant term from the left side and gives us: x - 4.5 + 4.5 = -1.1 + 4.5, which simplifies to x = 3.4. And there you have it! We have successfully solved the equation and found that the value of x is 3.4. This step-by-step approach, involving simplification, combining like terms, and isolating the variable, is the key to solving any linear equation. By mastering these techniques, you can confidently tackle a wide range of algebraic problems.

After solving an equation, it's always a good practice to verify the solution. This ensures that the value we found for the variable is indeed correct and satisfies the original equation. To verify our solution, we substitute the value of x we found (x = 3.4) back into the original equation: 10x - 4.5 + 3x = 12x - 1.1. Substituting x = 3.4, we get: 10(3.4) - 4.5 + 3(3.4) = 12(3.4) - 1.1. Now, we perform the arithmetic operations on both sides of the equation. On the left side, we have: 10(3.4) = 34, 3(3.4) = 10.2. So, the left side becomes: 34 - 4.5 + 10.2 = 39.7. On the right side, we have: 12(3.4) = 40.8. So, the right side becomes: 40.8 - 1.1 = 39.7. Comparing both sides, we see that: 39. 7 = 39.7. Since both sides of the equation are equal when x = 3.4, we can confidently conclude that our solution is correct. Verification is a crucial step in the problem-solving process. It not only confirms the accuracy of our solution but also helps us identify any potential errors in our calculations. By developing the habit of verifying solutions, we can build confidence in our mathematical abilities and ensure that we arrive at the correct answers.

While solving linear equations, it's easy to fall prey to common mistakes that can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them and ensure accuracy. One common mistake is failing to distribute properly. For example, if you have an expression like 2(x + 3), you need to multiply both the 'x' and the '3' by 2. Incorrectly distributing would lead to the wrong simplification. Another common mistake is combining unlike terms. Remember, you can only add or subtract terms that have the same variable raised to the same power. For instance, you can combine 3x and 5x, but you cannot combine 3x and 5x². Ignoring the signs of the terms is another frequent error. When moving terms from one side of the equation to the other, remember to change their signs. For example, if you have x + 5 = 10, subtracting 5 from both sides gives you x = 5, not x = 15. Careless arithmetic is also a significant source of errors. Even a small mistake in addition, subtraction, multiplication, or division can throw off the entire solution. It's crucial to double-check your calculations to minimize these errors. Finally, failing to verify the solution is a mistake that can cost you points on a test or lead to incorrect conclusions in real-world applications. Always substitute your solution back into the original equation to ensure that it satisfies the equation. By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and confidence in solving linear equations.

In this comprehensive guide, we've explored the intricacies of solving linear equations, using the equation 10x - 4.5 + 3x = 12x - 1.1 as our example. We've broken down the process into manageable steps, from simplifying the equation to isolating the variable and verifying the solution. We've also highlighted common mistakes to avoid, empowering you to approach linear equations with confidence and accuracy. Mastering linear equations is not just about finding the value of 'x'; it's about developing a logical and systematic approach to problem-solving. The techniques we've discussed can be applied to a wide range of mathematical problems and real-world scenarios. From calculating the trajectory of a projectile to balancing a budget, linear equations provide a powerful tool for understanding and solving the challenges we face every day. So, continue to practice and hone your skills in solving linear equations. The more you practice, the more proficient you'll become, and the more you'll appreciate the elegance and power of mathematics. Remember, every equation is a puzzle waiting to be solved, and with the right tools and techniques, you can unlock its secrets and reveal the hidden solutions.

x = 3.4